Marouane Fellaini's second-half header sent Everton into the last 16 of the FA Cup as they came from behind to beat Fulham at Goodison Park.
Danny Murphy handed Fulham an early lead when he slotted home from the spot after Johnny Heitinga was punished for a handball in the area.
However, Denis Stracqualursi put Everton level with a deft header from Landon Donovan's cross soon after.
Fellaini grabbed a deserved winner as he headed home another Donovan cross.
Everton had never beaten Fulham in the FA Cup before this game with the Cottagers having won all four of their previous meetings in the competition.
did you know?
Fulham have not won at Goodison Park since an FA Cup victory back in 1975
However, Martin Jol's side have been woeful on the road this season - their only away league victory came back in October at Wigan.
Despite that, it was the visitors who got off to the better start as Everton defender Heitinga was adjudged to have handled in the area.
The Dutchman slid in to block Chris Baird's strike and the ball came off his arm, with referee Howard Webb pointing to the spot without delay.
Murphy stepped up to send Tim Howard the wrong way with his calmly taken penalty and grab his sixth goal of the season.
Everton were incensed by the penalty decision but going behind jolted them into life and they should have equalised soon after when Shane Duffy headed wide from close range after he rose to meet Magaye Gueye's cross.
Tim Cahill, who scored his first goal since 2010 on Saturday against Blackburn, was denied a second of the season when David Stockdale made a superb save to push away his headed attempt.
Stephen Kelly was then the hero when he headed a Duffy shot off the line, but Everton's pressure soon paid off.
Donovan made space down the right-hand side before crossing for Stracqualursi, whose deft header gave Stockdale no chance.
did you know?
This was the eighth time in four seasons that Everton have had to face fellow Premier-League opposition in the FA Cup
Bryan Ruiz had Fulham's best chance before the break to put the Cottagers back in front but he hit a tame effort straight at Howard when there were better options on offer.
Everton pushed forward after the break but Stockdale was alert again to deny Stracqualursi, the Fulham goalkeeper diving at his feet to take the ball.
Andy Johnson was making his first start of 2012 but his first real action came when he hit a tame shot straight at Howard from inside the area just after the hour mark.
The game seemed to be petering out into a draw, but Donovan had other ideas as he turned provider once again.
Phil Neville played him in down the right and he sent in a perfectly placed cross for Fellaini to head back across goal and past Stockdale.
Jol introduced Bobby Zamora in a desperate attempt to earn a replay but even the England international could not change the course of the tie with Ruiz wasting their best chance to grab a dramatic equaliser when he fired a volley wide from inside the area.
Everton assistant manager Steve Round:
"We just about deserved it, we had the better chances.
"The penalty came against the run of play and it was a difficult one for the referee to give. It hit Heitinga's hand but we need some clarification on what is handball, and what isn't. There is a real inconsistency there.
"The team showed great sprit to come back and win it and every time we put the ball in the box we looked a threat.
"I'm really pleased for Denis Stracqualursi, he works so hard and he just needs to get the breaks.
"We're trying very hard to add to the squad before the transfer deadline but there is no real news at the moment."
Fulham manager Martin Jol:
"If you score a goal that early, you hope for more. You hope to go to the next round.
"But you have to defend well because you know the quality they have from crosses and midfield players.
"They had a strong spell of five corners in about two minutes, that is their strength. Under pressure you have to play around them and we couldn't do that most of the time.
"The result is disappointing but overall I think Everton deserved it."
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