The Premier League rivals played out a dull goalless draw in the original tie at the Stadium of Light to force a third-round replay, and goals from Sam Vokes and Andre Gray sealed the home side's place in the hat for the next round.
Burnley, whose record at home this season has been bettered only by Tottenham Hotspur and Chelsea in the Premier League, began brightly and could have taken the lead after just four minutes when Vokes nodded narrowly wide from Jeff Hendrick's cross.
Joey Barton, who scored the first goal of his second spell at the club to fire them to victory over Southampton at the weekend, was the next to come close when his long-range effort was parried away by Vito Mannone.
It continued to be the hosts who looked most like breaking the deadlock in the opening exchanges, and George Boyd fired another strike narrowly over as Sunderland hung on.
The Black Cats did manage to stem the tide once they had settled into the match, but they were unable to prevent the Clarets from taking the lead right on the stroke of half time when Vokes rose highest in the area to nod Tendayi Darikwa's cross home.
Sunderland, who were beaten 4-1 at Turf Moor in the Premier League on New Year's Eve, looked to make a quick start to the second half to haul themselves back into the match, with Sebastian Larsson sending a shot narrowly over shortly after the restart.
Burnley were soon back on the front foot, though, and Vokes and Boyd both came close to a second in quick succession just before the hour mark.
David Moyes looked to Jermain Defoe off the bench in an attempt to rescue his side's FA Cup hopes, but the two-time winners saw their fate sealed with just three minutes of normal time remaining when Gray latched on to a knockdown by Boyd before slotting his finish past Mannone.
There was no way back for the visitors in the closing stages despite Nick Pope being forced into a late save from a free kick as Sunderland slumped to a sixth consecutive match without victory and a fifth away defeat on the bounce for the first time since April 2014.
Burnley, meanwhile, have now won their last five home outings, while their clean sheet means that they have gone three games without conceding for the first time this season.
The Clarets will face Championship side Bristol City in the fourth round on January 28 after they beat Fleetwood Town in a replay.