Luis Sinisterra's first Premier League goal for Leeds cancelled out Anthony Gordon's first-half finish to keep Everton winless in the league this season after a chaotic 1-1 draw at Elland Road.
The away side took a 17th-minute lead through Chelsea target Gordon, who calmly finished past Leeds goalkeeper Illan Meslier after running on to Alex Iwobi's through ball.
But Leeds equalised 10 minutes into the second half thanks to Sinisterra's low drive from outside the box which wrong-footed a motionless Jordan Pickford on the Colombian's first Premier League start since joining from Feyenoord in July for £25.4m.
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Pickford came to the Toffees' rescue in the 74th minute as he superbly denied substitute Joe Gelhardt when through one-on-one.
Frank Lampard's side went close to claiming their first league win of the season as Rasmus Kristensen blocked Amadou Onana's close-range effort in the 79th minute and then four minutes later Meslier denied Nathan Patterson when put through by Gordon.
The Frenchman's save keeps Leeds unbeaten at home as they stay fifth while Everton got their third draw of the season to move out of the bottom three and up to 15th.
How Everton came close to their first league win of the season
Everton weathered early pressure from Leeds to strike first with their only shot on target of the opening 45 minutes.
Iwobi picked out Gordon's darting run with a through-ball that Leeds defender Diego Llorente failed to deal with, and the England U21 international showed why Chelsea are prepared to pay up to £60m for him with a low shot under Meslier for his second goal in successive matches.
Leeds were dealt a huge injury blow in the 30th minute when Rodrigo was substituted with a dislocated shoulder following a collision with Pickford with the Spaniard set for a scan on the injury.
His replacement Gelhardt had a hand in the Whites' second-half equaliser as he set free Brendan Aaronson, who found Sinisterra with the 23-year-old striking home from distance for his second Leeds goal after scoring in last week's Carabao Cup win against Barnsley.
Tempers then boiled over when Gordon and Kristensen went head to head as both players received yellow cards as a melee broke out on the touchline between the two benches.
Everton thought they had regained the lead through Demarai Gray's low finish, but it was ruled out for offside with VAR confirming the linesman's decision in a close call.
Leeds then came close to leading through Gelhardt but he was denied by Pickford's boot in a smart save by the goalkeeper.
Pickford's opposite number Meslier was also in good form as he stopped Patterson late on in a frenetic second half.
Marsch: Everton's time-wasting was terrible
Leeds head coach Jesse Marsch speaking to BT Sport about perceived time-wasting throughout the game from Everton:
"Yeah that was terrible. This is classic. For me, you can see when the players from Everton are kicking the ball away in the first minute of the game, it's clear they wanted to take the energy out of Elland Road.
"This is still an entertainment business and I wish the referee had handled that a little bit better and created more urgency.
"After it was 1-1 it became entertaining, but it was too late. The referee had the chance to affect the game and did not take advantage of that."
Lampard: Leeds' obsession with time-wasting was strange
Everton manager Frank Lampard speaking to BT Sport about Marsch's time-wasting complaints:
"I think they [Leeds] became obsessed with the time.
"We went 1-0 up and there were a couple of times the ball was in the crowd still and their bench was talking about time. I don't know if they want Jordan [Pickford] to sprint into the crowd and get the ball back and put it down. I didn't get it. I think it became an obsession.
"Then the crowd get on it and then they try to put pressure on the referee. I understand it. Goodison Park can be a tough place to come.
"I think it became a side issue because there were a lot of times there when we had to just go and get the ball. What do you want us to do? Strange."
Man of the match: Luis Sinisterra
Luis Sinisterra impressed on his first Premier League start for Leeds as he was a constant threat on the right side.
He was always part of their brightest moments and deserved his goal as he looks to be a smart summer signing from Feyenoord.
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Opta stats: Leeds commit most fouls in PL this season
- Leeds have conceded more fouls than any other side in the Premier League this season (65), committing at least 10 in each of their five games so far (10 tonight vs Everton).
- Leeds have earned eight points from their last four home league games (W2 D2), as many as they had from their previous 11 at Elland Road (W2 D2 L7).
- Since Frank Lampard's first Premier League game in charge of Everton in February, only Leicester (16) and Wolves (14) have dropped more points from winning positions than the Toffees (13).
What's next?
Everton host Liverpool in the Merseyside derby on Saturday at 12.30pm while Leeds travel to Brentford on the same day at 3pm.