Hugo Rodallega gave the Londoners a first-half lead just before the half-hour mark until an unfortunate own goal from the otherwise impressive Bettinelli was followed up by a fine strike from Ricardo Alvarez and Jordi Gomez’s stoppage-time penalty as Sunderland secured a 3-1 win.
A trip to giant-killing Bradford now awaits in the fifth round for Sunderland, who dominated proceedings for the most part.
However, poor finishing and fine goalkeeping from Bettinelli allowed Fulham to make the breakthrough in the 28th minute as Rodallega struck home their first real chance .
Ross McCormack threatened to add to that strike but Sunderland pushed for a leveller which came in slapstick fashion as Bettinelli somehow fumbled a deflected Patrick van Aanholt cross over the line.
It proved the catalyst to a Sunderland comeback and Alvarez, the evening’s stand-out player, cut in from the right to unleash a fierce drive to put the visitors ahead with 15 minutes remaining.
Gomez’s stoppage-time penalty rubber-stamped their progress to the last 16.
Steven Fletcher, a late inclusion in the line-up after Connor Wickham dropped out, threatened early on, finding the net only for the goal to be chalked off.
Kit Symons’ side were being reduced to counter-attacks and long punts forwards until the 27th minute, when a last-ditch intervention was all that prevented a Kostas Stafylidis cross being turned home.
The resulting corner caused confusion and Rodallega capitalised, firing home from close range.
McCormack felt he should have had a penalty moments later, only for referee Paul Tierney to wave away his appeals against Alvarez before the Scot missed the target before the break.
Jermain Defoe was proving a nuisance after the break and Alvarez threatened. Alex Kacaniklic headed wide at the other end, but that was a rare moment of respite for Fulham, who conceded in embarrassing circumstances as Bettinelli fumbled a Van Aanholt cross over his own line.
Seko Fofana fizzed an effort wide minutes after wasting a fine chance to shoot, before Bettinelli raced off his line to deny Alvarez.
The Inter Milan loanee was not to be denied for long, though, running into the box and cutting onto his left foot, before belting a strike home.
Giaccherini raised his hands to Fofana – an incident which led both players to be booked – and the Italian was lucky not to see red.
Fulham made late changes in a bid to force the match into extra time, but there was to be no leveller. And Sunderland pulled further ahead in stoppage time, with Gomez scoring from the spot after Shaun Hutchinson fouled Danny Graham.