Late goals from Ricardo Alvarez - his first for the club - and Jordi Gomez completed a Sunderland comeback on Tuesday, ensuring the Premier League side will now face Bradford City, the conquerors of Chelsea, in round five.
Fulham took the lead through Hugo Rodallega following early pressure from the visitors, but were then left to rue an error from Bettinelli, who dropped a seemingly innocuous cross from Patrick van Aanholt into his own net under minimal pressure.
Gus Poyet's men took full advantage, Alvarez making it 2-1 with a fine individual goal in the 75th minute.
And any hopes of a Fulham fightback were ended in stoppage time when Gomez converted a penalty awarded for a foul on substitute Danny Graham.
Sunderland were forced to make a late change to their side, Steven Fletcher replacing Connor Wickham due to the latter picking up an injury in the warm-up.
Yet that didn't seem to affect the visitors as they made an assured start, and Bettinelli had to make instinctive early saves to deny Gomez and Fletcher, with the Scotland forward also seeing a goal ruled out for offside.
The hosts had been on a four-match unbeaten run up until their 2-1 league defeat at Blackburn Rovers last time out, but they struggled to build any momentum in the early stages.
Nevertheless, it was Fulham who struck first when Sunderland's defence failed to clear a corner. Cauley Woodrow saw his effort blocked on the line, but Rodallega was on hand to dispatch his 10th goal of the season from close range.
Buoyed by that goal, Kit Symons' side could have been awarded a penalty just four minutes later when Alvarez brought down Ross McCormack, but referee Paul Tierney waved away the appeals.
The Championship outfit continued to threaten as half-time drew closer, McCormack almost doubling their advantage when he fizzed an effort just wide.
Fulham also made a bright start after the interval, yet Sunderland were gifted an equaliser when Bettinelli flapped haplessly at Van Aanholt's deflected cross and allowed the ball to drop over the goalline.
Emanuele Giaccherini curled a sumptuous effort narrowly off target as Fulham struggled to come to terms with the setback.
Manchester City loanee Seko Fofana provided encouragement for the home side, but it was Sunderland winger Alvarez who stole the show.
The 26-year-old, on loan at the Stadium of Light from Inter, cut in from the right flank - beating two defenders in the process - before unleashing an unstoppable drive that left Bettinelli helpless.
Victory was then put beyond all doubt for the visitors when Graham went down in the area under pressure from Jack Grimmer, and Gomez stepped up to dispatch coolly from the spot.
Elsewhere, Preston North End booked a fifth round home tie against Manchester United as they came from behind to beat Sheffield United 3-1 at Bramall Lane.
Paul Gallagher scored twice either side of Paul Huntington's header to cancel out Jamie Murphy's first-half effort and send his former club crashing out of their second cup competition in the space of a week.
The Scottish forward, who also netted in the first tie at Deepdale, levelled with an exquisite free-kick before wrapping up the mouth-watering clash with 11-time winners United.
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