Frank Lampard says it was a "big deal" for Everton to beat West Ham 1-0 to secure their first Premier League win of the season.
Neal Maupay opened his Everton account with a half-volley from just inside the area in front of the Gwladys Street End (53) to the delight of the home crowd, who had endured a drab first half.
There was a scare for Everton as West Ham substitute Said Benrahma's deflected left-foot shot struck the post in the second half, but they held on for consecutive league clean sheets for the first time in over a year.
"At this moment it is massive for us to get a win," said Lampard. "Hopefully we will get better but to get a win today was a big deal for us."
West Ham produced another frustrating performance as they struggled to keep up with Everton's energy following Thursday night's 3-2 win over Danish side Silkeborg in the Europa Conference League and remain 18th in the table.
"We weren't very good," Moyes told Sky Sports. "Today we found out little bits [about the new signings], but it was mainly the players I know about - they are the ones who let me down with poor play overall."
How Maupay ignited Everton's season
Both sides came into this game knowing a defeat would see them spend the next two weeks in the relegation zone during the international break, but, despite their best efforts, the first half was characterised by a distinct lack of quality in the final third.
West Ham created the best opening as Jarrod Bowen's cut-back looked certain to set up Michail Antonio in the six-yard box only for Conor Coady to produce a goal-saving block in what was the visitors' only moment of real threat before the break, while the hosts failed to hit the target.
The match required a goal to bring a much-needed lift to proceedings and Maupay delivered emphatically as he received Alex Iwobi's pass on the edge of the box and produced a quick turn and shot on the bounce to leave West Ham goalkeeper Fabianski helpless.
Jarrod Bowen's low, drilled shot from just inside the box was kept out well by Everton's James Tarkwoski, who also got a vital touch on Benrahma's long-range effort to take the ball onto the post with just over 15 minutes to go.
Bowen had another chance but fired over from around the penalty spot after Pablo Fornals' cut-back, while Tomas Soucek headed narrowly wide from a corner as West Ham failed to make the most of their 14 attempts.
Substitute Maxwel Cornet had the best opportunity to equalise in the closing stages after a driving run by West Ham captain Declan Rice, only for his touch to let him down when one-on-one with Everton's stand-in goalkeeper Asmir Beogvic, ensuring the Hammers make it three league games without a win and stay on four points.
Moyes: The players I know are letting me down
West Ham boss David Moyes speaking to Sky Sports:
"We weren't very good. We picked up a bit after they scored. I think the first half was a poor game all round really, the amount of times both teams turned the ball over.
"We had a game on Thursday night and you have to take that into consideration as well. We are also putting new players in and trying to find out more about them. Today we found out little bits, but it was mainly the players I know about - they are the ones who let me down with poor play overall.
"We are showing signs that we are getting back to it, we didn't do it today. We couldn't put three passes together, and when we got anything in the final third it was almost always the wrong decision.
"In the last 20 minutes we had some chances to at worst get an equaliser, so if you really analyse the end of it we missed a few big chances."
Lampard: This is a massive win for us
Everton manager Frank Lampard:
"It's really important because - obvious fact but - when you play well and don't get what you think you deserve it leaves the table not looking so great.
"For the confidence of the group to see the work they are doing and the spirit they have got can bring some results in terms of wins.
"We are starting to fix some things, we are harder to beat, we don't concede so much. It is great for the group, particularly going into the break, because it gives confidence for the group and club alike.
"It is a win we feel we have deserved in house.
"There is no doubting that because of how we have been playing and the feeling in the group. The feeling I have is the spirit is growing and the application of the group is big but you need a result.
"The fact is we are not conceding so much and are unbeaten in four in the league coming into this, but we all know we want wins. We are 13th now and there is a real hope we can we move in an upward direction.
"There is so much work to do on that front. At this moment it is massive for us to get a win. Hopefully we will get better but to get a win today was a big deal for us."
Maupay: I needed that goal
Everton striker Neal Maupay speaking to Sky Sports:
"I needed this goal because I haven't been playing much in recent months; in the end at Brighton I wasn't playing and then I came here pretty late.
"This was only my second game of the season, I needed this. I know everyone at the club wants me to do well and score goals.
"I'm here to help and I trust my team-mates to put me in positions to score. Today I got the rewards."
Iwobi and Gray lead the way for Everton
Neal Maupay grabbed the headlines with his match-winning goal on Sunday, but after the game his comments highlighted the importance of those around him.
"We've got good players in the midfield and on the wing and I trust them to make the difference," the striker told Sky Sports.
It was Alex Iwobi who delivered the precise pass for Maupay to score, opening his body up quickly before drilling it into the Frenchman's feet without hesitation.
Iwobi has rediscovered his confidence under Frank Lampard and is becoming a key player for Everton as a No 10 or at wing-back. He carried the ball well in central areas on Sunday and was impressive in his distribution as showcased by the assist.
No Everton player had as many touches, entries into the final third or passes in the final third on Sunday. Iwobi also had a team-high two chances created.
The only man to equal that stat was Demarai Gray, who could have had an assist or two himself.
Gray provided a different kind of threat out wide. The former Leicester winger was always asking questions of Vladmir Coufal with his driving runs and one of the few players who injected some life into the game.
The 26-year-old delivered more crosses than anyone on the pitch (11) and was unfortunate his team-mates weren't gambling at the back post on a couple of ocassions.
His fearlessness is infectious and lifts the rest of the team. Everton will rely on him for that spark as they go through this transitional period, with more frustrating afternoons expected.
Worrying times for West Ham
West Ham look out of ideas after seven Premier League games. Five defeats, one win and a draw mean they go into the international break in the bottom three.
Few would have expected that after last season, especially after a summer which saw them spend £179m on new signings. Only Chelsea and Manchester United spent more in the Premier League.
Sunday's defeat to Everton was another Hammers performance below the standard we have become accustomed to under David Moyes. They had the chances to equalise at the end, although there was never really a feeling that they had the conviction to score
Moyes called out the players that brought him so much success last season after the defeat to his former club. "Today we found out little bits [about the new signings], but it was mainly the players I know about - they are the ones who let me down with poor play overall," he told Sky Sports.
Only two of the six new first-team signings were involved from the start on Sunday. Moyes has been holding them back as he gives the arrivals time to gel with their new team-mates. But maybe it's time to roll the dice?
England World Cup Squad watch
There was an opportunity for Jarrod Bowen to repay the faith Gareth Southgate put in him by sticking with him in his latest England squad despite an unconvincing start to the season.
But Bowen had another frustrating afternoon for West Ham at Everton. He's managed just two assists this season - both coming in the Europa League Conference League.
Bowen was kept quiet by Everton's impressive centre-back pairing of Conor Coady and James Tarkowski, who have given Frank Lampard's side a solid base from which to build this season.
They could be ones to watch for Southgate's World Cup squad, but likely only as back up if selected.
West Ham captain Declan Rice kept things ticking over for his side on Sunday and produced a characteristically solid performance. He is certain to be a starter for Southgate.
The deadline for Gareth Southgate to name his squad is October 19. The World Cup begins on November 20, with England's first game against Iran on November 21.
What's next?
After the international break, Everton travel to Southampton (3pm), while West Ham host Wolverhampton Wanderers live on Sky Sports Premier League (5.30pm), with both games taking place on Saturday October 1.