The striker, who was playing for non-league side Yeading in the competition last year, scored two goals to cancel out Julio Arca's lucky effort.
The brace puts the League One side in the hat for the fifth round of the competition and puts more pressure on Sunderland boss Mick McCarthy.
It was Brentford who enjoyed the better of the first half and they were starting to curse themselves for failing to capitalise on Sunderland's amateur defending.
The Premiership side looked all at sea at the back with Gary Breen and Danny Collins particularly guilty of losing their markers at set-pieces.
Isaiah Rankin set the tone for the afternoon in only the fifth minute when his shot from 18 yards narrowly fizzed past Kelvin Davis' post.
Campbell then blazed over before Davis was called upon to tip Sam Sodje's looping header over the bar.
The League One side were so dominant that Sunderland had to wait until the 33rd minute before they mustered their first shot on target, but Dean Whitehead's half volley was straight into Stuart Nelson's arms.
Whitehead then went close for Sunderland just before the break, but his shot was well saved by Nelson.
Brentford started the second half with a flourish. Jay Tabb's shot was deflected into Campbell's path but the striker's shot was off target.
It was Campbell again causing Sunderland problems and he was unlucky to see his shot from distance swerve off target.
Sunderland had their chances too. Jon Stead could not find the killer touch to delft the ball past Nelson in the 54th minute.
Sodje then made a crucial tackle to deny Kevin Kyle a shot on goal on his return after an 18-month absence following a hip operation.
But just as Sunderland thought they were getting into the game Campbell struck in the 56th minute.
Tabb's long ball looked hopeful but somehow the scrawny striker outpaced and out muscled Neill Collins and Breen before rounding Davis and calmly slotting the ball home.
It was a fluke goal from Arca in the 66th minute which got the Premiership strugglers back into the game.
The winger was wide on the left flank and curled the ball hopefully into the box, only for it to dip under Nelson's crossbar.
But their joy was turned to misery when Campbell added his second of the afternoon with just two minutes remaining on the clock.
The livewire striker latched on to Lloyd Owusu's knockdown and raced around Breen before slotting the ball past Davis his 12th goal of the season.
The strike sent the Brentford fans delirious which was followed by a pitch invasion at the end.