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.
Jamie Paterson had headed Forest into a 28th-minute lead, but United - the lowest-ranked team left in the competition - hit back through Conor Coady after the break following a shocking Dorus de Vries error.
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.
The hosts did not seem overawed by their Championship opposition and had a chance to take a 12th-minute lead when John Brayford cut in from the left, but struck straight at De Vries.
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.
Forest controlled possession, but United were looking dangerous on the break and Jose Baxter wasted a great chance on 25 minutes, hitting a tame shot straight at De Vries.
United were made to pay for their profligacy three minutes later when Andy Reid crossed from the left and Paterson did brilliantly to steer the ball across Mark Howard and in at the far post.
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.
If the United players needed any further motivation at half-time, the news from the draw of a potential quarter-final tie with Wednesday should have done the trick.
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.
United continued to press and, with clear cut chances hard to come by, got a huge slice of luck when De Vries spilled a routine Brayford cross to present Coady with an open goal and the youngster made no mistake.
The tie looked to be heading for a replay, but Greg Halford was penalised for handball inside the area on 90 minutes to give substitute Porter the chance to win the game from the penalty spot.
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.
