Porter kept his cool to convert a 90th-minute penalty and then netted again two minutes later to settle the fifth round tie at Bramall Lane on Sunday as United came from behind to claim another FA Cup scalp.
Having already dumped out Fulham and Aston Villa, United manager Nigel Clough will savour this victory over his former club and counterpart Billy Davies, with no love lost between the duo.
A replay at the City Ground looked on the cards before Porter's late double proved decisive and sent United through to the last eight, where the League One strugglers will play host to either Sheffield Wednesday or Charlton Athletic.
A spell of Forest pressure in the 20th minute ended with Simon Cox appearing to be felled by Harry Maguire inside the penalty area, but Forest's surprisingly half-hearted penalty appeals were waved away.
Given the two clubs' respective league positions, United might have been expected to crumble, but the home side continued to threaten and wasted another good opportunity when Stefan Scougall's weak shot was saved after he latched onto Baxter's throughball.
And Clough's men did seem to return from the dressing room with renewed vigour, putting Forest on the back foot and forcing two corners in quick succession with Maguire's attempt deflected over before Scougall's half-volley was well-held by De Vries.
The former Derby County man converted easily and, as Forest went in search of an equaliser, United hit them on the break, Porter finishing off Jamie Murphy's right-wing cross to seal a famous victory.
