Danny Murphy gave Fulham a controversial early lead from the penalty spot at Goodison Park on Friday night but hard-working Everton were rewarded for determined fightback.
Landon Donovan was twice the provider as Stracqualursi, with his first goal for the club, equalised before half-time and Fellaini settled the contest 17 minutes from time.
Stracqualursi, who has made limited impact since joining Everton on a season-long loan from Tigre, had been a surprise inclusion for Everton ahead of Louis Saha.
Fulham also made a change in attack as Andrew Johnson started ahead of Bobby Zamora, who was left on the bench. Everton kept faith with 20-year-old Shane Duffy at the back and gave Magaye Gueye a chance while Fulham bolstered their defence with Aaron Hughes and Chris Baird.
Everton controlled the early possession but it was Fulham who claimed the lead after the first move of note in the 13th minute.
It also came in contentious circumstances as John Heitinga was adjudged to have handled as he slid in to block a Damien Duff shot.
John Arne Riise had got behind the defence as he met a superb cross-field pass from Murphy and cut the ball back for Duff in front of goal.
Duff shot instantly and the sliding Heitinga could do little to get his arm out of the way from point-blank range, although it was raised above his head.
Former Liverpool midfielder Murphy calmly stroked home the penalty but Everton, feeling wronged by referee Howard Webb's decision, responded well to the setback.
They won a succession of corners and caused problems from each of them with Duffy first heading a good chance wide from a Gueye cross.
Cahill then forced David Stockdale to save with a back post header from a Donovan delivery.
The next corner created even more difficulty for Fulham as Stockdale lost the ball and Duffy turned to curl a shot back at goal. Hughes came to the rescue as he dived across to head behind.
Everton's reward for this good spell came from open play as Stracqualursi headed them level after 27 minutes.
The Argentinian was making only his third start for the Toffees but justified his selection as he met a Donovan cross with a firm header that looped beyond the reach of Stockdale.
Despite their long spell on the front foot, Everton were almost caught out when Murphy tricked them by playing a free-kick across the field instead of lofting it into the box.
Baird struck team-mate Bryan Ruiz as he tried his luck from long range and the Costa Rican hit a snap-shot straight Tim Howard.
Stockdale remained the busier keeper and produced a fine save to keep out a powerful long-range shot from Donovan.
Everton went close again just before the break as Cahill fired wide in an injury-time goalmouth scramble.
Fulham had the first chance of the second half when Clint Dempsey, who has scored two hat-tricks in his last three games, headed over from a Ruiz cross.
Cahill was booked when he slid in from behind on Murphy, who required treatment before continuing.
The game went quiet for a spell as both sides cancelled each other out before Everton roused themselves for a late rally.
A promising Everton break came to an end when Darron Gibson, looking for Gueye at the far post, hit a cross far too hard.
Donovan then drove another shot across goal from distance.
Everton made the decisive breakthrough as Donovan hit a deep cross to the far post and Fellaini guided a fine header back across Stockdale into the top corner.
Stockdale stretched to try to claw it out but Fellaini's placement for his third goal of the season was perfect.
Everton almost made the game safe with the next attack as Cahill thundered a header against the bar from a Phil Neville cross.
The ball then hit the bar again as Hughes attempted to head clear from the rebound but this time it went over.
Fulham boss Martin Jol sent on Zamora for Duff in attempt to change the momentum of the game with just over 10 minutes remaining.
It failed to have the required effect for the Londoners and Everton played out time to advance to the last 16.