Thomas Gravesen lifted the gloom over Everton with two crucial goals to seal victory at Crystal Palace.
Mark Hudson headed Palace's opener from fine work by winger Wayne Routledge.
But Gravesen quickly levelled from the spot after Palace keeper Julian Speroni fouled Kevin Campbell following his own poor clearance.
The Dane then hit a super second, and despite seeing Gary Naysmith sent off, Gravesen was able to slip Marcus Bent through for Everton's third late on.
It was harsh on Palace, but Everton manager David Moyes will have been delighted with the win after only one victory on their travels last term.
Palace celebrated the return of Premiership football to Selhurst Park by taking an early lead.
Wayne Routledge took advantage of poor defending by Naysmith to cross for the unmarked Hudson to head home at the far post.
And they should have doubled their lead five minutes later when Andy Johnson raced clear on to Dougie Freedman's pass.
Johnson rounded Nigel Martyn, but veteran defender Alan Stubbs rescued Everton with a brilliant goal-line clearance.
Everton took advantage of the reprieve to equalise in farcical circumstances in the 17th minute.
Speroni put Palace's defence under pressure with a poor clearance, and when he tried to retrieve his error he only succeeded in fouling Campbell.
Referee Mark Clattenburg awarded the spot-kick and Gravesen calmly fired home.
Palace could have regained the lead three minutes before the interval when Freedman hooked inches over the bar after Everton goalkeeper Nigel Martyn had misjudged a corner.
Bent almost marked his first Premiership start for Everton with a goal after 52 minutes, bringing a fine save from Speroni with a right-foot volley.
Everton then made a substitution, sending on Duncan Ferguson for Campbell.
And Moyes' side went ahead in the 62nd minute, courtesy of a magnificent strike by Gravesen.
The Dane controlled the ball and dummied before curling a finish past Speroni into the top corner from 20 yards.
Speroni made some amends for his earlier error with a crucial block from Bent.
The Merseysiders were reduced to 10 men with 19 minutes left when Naysmith received a second yellow card for senselessly dragging back Routledge.
Palace then had strong penalty appeals turned down when Johnson tumbled under challenge from Tony Hibbert.
And their agony increased when Everton broke clear to snatch a third after 82 minutes, Gravesen sending Bent clear to finish calmly past Speroni.
Crystal Palace: Speroni, Boyce (Kaviedes 77), Hudson, Popovic,
Granville, Routledge, Riihilahti (Derry 73), Hall, Kolkka,
Subs Not Used: Hughes, Borrowdale, Kiraly.
Goals: Hudson 9.
Everton: Martyn, Hibbert, Stubbs, Naysmith, Carsley,
Osman (Pistone 75), Yobo, Gravesen, Kilbane,
Campbell (Ferguson 54), Bent (Watson 85).
Subs Not Used: Wright, McFadden.
Sent Off: Naysmith (71).
Goals: Gravesen 19 pen, 62, Bent 82.
Ref: M Clattenburg (Co Durham).