Colin Calderwood's side looked likely to join the list of giant-killers in the competition, as they held onto the lead given to them by Robert Earnshaw's 60th-minute free-kick.
But Phil Bardsley flicked the Black Cats level four minutes before the end of normal time - and substitute David Healy struck early in extra-time to see Sunderland through 2-1 and ensure Keane enjoyed his visit to the City Ground, where he made his name as a player.
Sunderland had begun the match with purpose and came close to taking a fourth-minute lead when Danny Collins glanced a header wide from a corner.
James Perch responded with a crisp, rising shot that cleared the bar at the other end and Matt Thornhill should have done better when a looping Julian Bennett cross picked him out perfectly - but he headed over from eight yards.
It took an outstanding touch from Paul Smith to keep the match deadlocked. Djibril Cisse burst into sufficient space to strike a low shot, which the keeper pushed acrobatically onto the post.
Smith had to be on his toes again when Dean Whitehead was sent scampering clean through by substitute Liam Miller, with the keeper spreading himself well to block his crisply struck drive.
Forest themselves spurned two further chances, with Chris Cohen failing to properly connect after getting behind the Sunderland defence and Earnshaw failing to make any contact with a clever Luke Chambers cross.
But the striker made no mistake in the second half when, having been felled by Daryl Murphy on the edge of the Sunderland box, he regained his feet to smash an unstoppable 20-yard free-kick into the corner of the net on the hour mark.
Earnshaw could have had another after dropping his shoulder to elude two defenders in the box, only to rifle his shot over the bar from 12 yards.
And former Forest player Andy Reid spurned the chance to silence the home fans, who had taunted him over his weight, when he ballooned a fine chance into the side-netting.
But extra-time was forced when Smith brilliantly parried a searing shot from Miller. However, the in-form shot-stopper could only look on as El Hadji Diouf drove the loose ball back into the danger zone, where Bardsley stuck out a boot to deflect the ball inside the post.
And it took only three minutes of the additional period for Sunderland to take the lead, with Smith making another remarkable save to deny Grant Leadbitter as he raced into the box, only for fellow substitute Healy to stroke him the rebound.
Chambers thumped a late header against the bar as Forest looked for an equaliser but Sunderland were generally comfortable as they held on to their lead.