New head coach Francesco Guidolin enjoyed a winning start as Swansea registered back-to-back Barclays Premier League wins for the first time this season with a 2-1 victory at Everton.
The Italian watched Monday's win over Watford from the stands and it is a matter for debate how much he influenced this result - the Swans' first ever league triumph in the 22 meetings between the two - as the visitors did not have to play particularly well to beat the worst home defence in the league.
Roberto Martinez's side have now conceded 22 goals in 12 home games, three more than anyone else, and the ease with which a Swansea team which had not scored in the first half in their previous eight away league matches was a concern.
A sluggish Everton never got going and gifted Swansea an the opener when John Stones' woeful backpass led to Gylfi Sigurdsson's penalty.
A Jack Cork own goal, denying Gareth Barry his first strike for 88 matches, was quickly followed by Andre Ayew's strike - and that was to prove decisive.
With captain Phil Jagielka rested ahead of Wednesday's Capital One Cup semi-final second leg, Stones was the senior centre-back - but he had a match to forget as his tendency to overplay proved costly.
That contrasted sharply with his opposite number Ashley Williams, whose old-fashioned approach saw him take no chances - or prisoners - and keep in-form Toffees striker Romelu Lukaku relatively quiet.
Everton's afternoon began badly when they lost Muhamed Besic, who had hit the post with the first shot of the match, to injury after just 10 minutes.
But that was only the start as in the 18th minute Stones delayed too long before leaving a square backpass short, which meant Tim Howard's attempted clearance only hacked down Ayew and Sigurdsson hammered home the penalty.
Such is the regard Howard is currently held in by many supporters at least one Everton fan called for his own goalkeeper to be sent off for the foul but however slow the American's reactions were Stones had to take his fair share of the blame.
However, the hosts fashioned a well-taken 27th-minute equaliser when Gerard Deulofeu's corner was backheeled by Barry via Jack Cork and in off the far post.
The celebrations were short-lived as injury to Kevin Mirallas forced a second substitution before Swansea went ahead again.
Deulofeu thought he should have had a handball decision against Williams but the visitors countered with Neil Taylor going past makeshift right-back Bryan Oviedo to set up Ayew, whose shot deflected off Stones past Howard.
After the break Deulofeu finally got into the game with a series of viciously whipped-in crosses but Lukaku, in particular, could not take advantage.
Guidolin, adopting a very Italian approach, was taking no chances and sent on defender Jordi Amat for winger Wayne Routledge and Eder for Ayew to shore things up.
Substitute Seamus Coleman rolled a late shot wide and then volleyed over from four yards with the last kick of the game but Everton hardly put Lukasz Fabianski under huge pressure as they chased the game.
The result lifted Swansea four points clear of the relegation zone and only four points behind Everton, in 12th.
TWEET OF THE MATCH
"The Everton fans have got their young players in a nervous state here!! Looks like they wont allow em time to make mistakes & learn...."
Rio Ferdinand @rioferdy5
Tim Howard: 6 (out of 10)
Bryan Oviedo: 5
John Stones: 5
Ramiro Funes Mori: 6
Leighton Baines: 6
Muhamed Besic: 5
Gareth Barry: 6
Gerard Deulofeu: 6
Ross Barkley: 6
Kevin Mirallas: 5
Romelu Lukaku: 5
Tom Cleverley (on for Besic, 11): 6
Steven Pienaar (on for Mirallas, 28): 6
Seamus Coleman (on for Oviedo, 67): 6
Lukasz Fabianski: 6
Angel Rangel: 6
Ashley Williams: 7
Federico Fernandez: 7
Neil Taylor: 7
Leon Britton: 7
Jack Cork: 6
Ki Sung-Yeung: 7
Wayne Routledge: 6
Gylfi Sigurdsson: 6
Andre Ayew: 7
Jordi Amat (on for Routledge, 77): 6
Eder (on for Ayew, 89): 5
Kyle Naughton (on for Rangel, 90): 5
Andre Ayew: Scored and his incessant running gave Everton centre-back John Stones in particular all kinds of problems.
MOMENT OF THE MATCH
Substitute Seamus Coleman missing an open goal from four yards with the last kick of the match which would have snatched the point Everton should at least have had.
VIEW FROM THE BENCH
Francesco Guidolin, often squatting down in technical area, may have had a game plan but even he could not have accounted for being gifted the opener. Moved to shore things up midway through the second half with some safety-first substitutions. Roberto Martinez was hamstrung by two early injuries and he cannot be blamed for a player missing an open goal from four yards but a lack of intensity suggests something was wrong from the start.
MOAN OF THE MATCH
A toss-up between Everton's leaky defence - who have the worst home record in the league - or the missed chances at the other end. Both particularly damning.
WHO'S UP NEXT?
Manchester City v Everton (Capital One Cup, Wednesday January 27)
West Brom v Swansea (Barclays Premier League, Tuesday February 2)
Match-winner Ayew was pleased to ensure a winning start for his new coach Guidolin but urged Swansea not to get carried away by their maiden league victory over Everton.
The Ghanaian told Sky Sports: "I'm happy for him, I'm happy especially for the supporters and the club, they have been behind us in this difficult moment. We need to continue like that. We have a victory here but we haven't done anything yet.
"There are a lot of games left. We have to work harder and I know we can even do better."
Captain Williams hopes the struggling Swans can now continue the momentum at West Brom in nine days' time after collecting six points from their last two matches.
He said: "It's about trying to put a run together - that's what we're speaking about now.
"Winning at a place like Goodison Park is massive for the club. We've not done it before. We know what it means to everyone involved, and we've got to keep picking up points every week."
Ayew felt being able to walk away with all three points was largely due to their resolve.
He added: "I think what was important was the team spirit that we showed. We showed a lot of determination. In the first half we played well. We got control of the ball till the last 10 minutes of the first half where they were putting us under pressure.
"The second half we knew was going to be difficult. We had a chance where Wayne (Routledge) could have killed it but the game was very difficult for the strikers.
"But I feel we showed a lot of desire and, when you show a lot of desire, sometimes the luck comes to your side."