Mick McCarthy's side made most of the running in the opening minutes but Hart's young soldiers seemed more equipped this term.
The ever-present Marlon Harewood opened the scoring in the 19th minute following a lovely midfield move.
Irish international Andrew Reid played a defence-splitting pass straight to Harewood's feet, he rounded Mart Poom and found the net with his shot.
In the 27th minute Forest had a good shout for a penalty turned down by the referee.
Sunderland defender Stephen Wright clearly obstructed David Johnson in what looked like a blatant penalty.
Forest's bad luck continued in the 36th minute when Harewood put the ball into the visitors net only for the referee to disallow the goal when video evidence clearly showed that Kevin Kyle clattered into his own keeper.
The second goal eventually came in the 40th minute.
McCarthy will have his defenders in early on Monday morning to show them the replay of this goal.
It was simple. Sunderland's defence fell asleep at Reid's corner and Mathieu Louis-Jean rose to head home at the near post with Poom stranded and Phil Babb rooted to his the post as the ball sailed home.
The second half lacked the pace and fire of the first mainly due to the heat which was a major factor in the game.
Hart was a victim of the heat as he seemed to collapse at the pitch side. After the game he said he was OK and that he just had heat exhaustion and a bad migraine.