A few minutes later Earnshaw atoned for that miss when he won a free-kick on the edge of the Sunderland box.
The Welsh forward produced a moment of class to curl it around the wall and past a stranded Craig Gordon.
Sunderland's players looked frustrated, while Keane fumed on the sidelines.
Ex-Forest favourite Andy Reid volleyed into the side-netting as time looked to be running out on the visitors' cup hopes.
Sunderland's leveller came somewhat out of the blue - and from an unusual source in Bardsley.
It was hard luck on the impressive Smith, who brilliantly kept out Liam Miller's shot but saw Diouf fire the ball back into the centre for Bardsley to score his first goal in English football.
Thornhill nearly restored Forest's lead in the last minute but headed off target when unmarked.
Three minutes into extra-time Cisse found the fresh legs of Leadbitter, who worked his way into a shooting position only to see his effort saved by Smith.
But Healy was in the right place to follow up and sidefoot the rebound into an empty net.
A tiring Forest had one last chance to take the game to a shoot-out but Chambers' header from Earnshaw's free-kick struck the bar.
Sunderland boss Roy Keane told BBC Radio Newcastle:
"It was a good Cup tie, credit to Forest, they made it very hard for us, but credit to my lads for getting us back in and getting us through.
"This time last year we were out of the Cup so plusses and minuses. The minus would be the injuries and the fact that we can play a lot better, the plusses would be that we showed good character, scored a great goal and we created chances in the first half."
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