Swansea recorded their first ever league win over Everton as they hung on to record a 2-1 success in Francesco Guidolin's first game as manager.
The visitors went in front through Gylfi Sigurdsson's penalty after Tim Howard brought down Andre Ayew following John Stones' weak backpass.
Gareth Barry's deflected flick made it 1-1 just eight minutes later but Ayew restored the advantage with an effort which looped in over Howard off Stones.
The hosts dominated the second half and a glaring injury-time miss from Seamus Coleman summed up their day as Swansea moved up to 15th in the Premier League table.
As well as a priceless three points for the new head coach, the result gives the team he has inherited back-to-back wins for the first time this season.
Swansea arrived on Merseyside with a win against their opponents in the Capital One Cup last season their only victory in 24 previous meetings.
Roberto Martinez's team began brightly and Muhamed Besic hit the post with a well-struck effort from distance in the opening moments.
However just as quickly, they lost him to injury as he had to be replaced by Tom Cleverley due to a recurrence of a hamstring problem.
Gerard Deulofeu was to play a more prominent role in the game as it wore on and he served an early warning of his intentions with a curling effort which Lukasz Fabianski held.
In response, Swansea went ahead - but they did so in somewhat farcical circumstances as Stones and Howard contributed in equal measure to a defensive disaster.
The England international's pass back to his goalkeeper was light and Ayew realised quickly, darting in to try and exploit his mistake.
Rather than react, Howard waited for the ball to come to him and succeeded only in bringing Ayew down, conceding a penalty and seeing Sigurdsson convert it with confidence and composure.
Everton were level eight minutes later, Barry scoring his first goal in two years with the help of a big deflection off Jack Cork which may yet be significant enough to mean it goes down as an own goal.
Rather than build on their swift comeback, Everton contrived to fall behind once more having also lost Kevin Mirallas to injury.
The ball appeared to strike Ashley Williams' hand in the build-up before Neil Taylor found Ayew, whose shot deflected off Stones and beyond Howard.
That made for a second half in which Swansea probably expected to be on the back foot and they weren't to be surprised but not before they came close to a 3-1 lead immediately after the restart - Wayne Routledge's effort being well smothered by Howard - but then Everton dictated.
Ross Barkley put the ball over from close range before realising Leighton Baines had been flagged offside and Deulofeu sent in a succession of wonderful crosses but none were finished off.
Romelu Lukaku - making his 100th Everton appearance and seeking a club record-equalling 16th Premier League goal of the season - was culpable.
Barkley too failed to take advantage of the terrific service Deulofeu provided and Mirallas' replacement Steven Pienaar was the same.
Lukaku then headed over, Coleman screwed a weak effort past the post when he should have done better and both Barry and Coleman claimed for penalties but neither were given.
Even when it seemed Everton had lost their chance to equalise, they had one more golden opportunity to do so.
But with the last kick of the match, Coleman again produced a poor finish at a corner and put the ball high over the bar when it seemed easier to score to condemn his team to a fifth successive Premier League game without a victory and a stream of boos from the Goodison terraces.
Player ratings
Everton: Howard (5), Baines (6), Stones (5), Funes Mori (6), Oviedo (6), Barry (6), Besic (4), Mirallas (4), Barkley (6), Deulofeu (8), Lukaku (6)
Subs used: Cleverley (6), Pienaar (5), Coleman (3)
Swansea: Fabianski (7), Rangel (6), Fernandez (7), Williams (7), Taylor (5), Britton (6), Cork (6), Ki (6), Sigurdsson (6), Routledge (7), Ayew (6)
Subs used: Naughton (1), Amat (1), Eder (2)
Man of the match: Gerard Deulofeu