Forest took a two-goal lead inside the opening four minutes, with Taiwo Awoniyi and Willy Boly on the scoresheet, but Christian Eriksen halved the deficit in the 17th minute.
Casemiro then levelled the scores early in the second period before Bruno Fernandes won it for the Red Devils from the penalty spot in the 76th minute, nine minutes after Joe Worrall had seen red for the visitors.
The result has moved the 20-time English champions into fifth position in the table on six points from three matches, while Forest are down in 14th, boasting three points from their opening three games.
Forest incredibly made the breakthrough in the second minute of the contest, and it was a terrific goal from Awoniyi, who broke clear following a Man United corner before sliding the ball into the back of the net under pressure from Marcus Rashford.
A bad start for the home side got considerably worse in the fourth minute, with Morgan Gibbs-White's free kick being converted into the back of the net by Boly.
Fernandes tested Forest goalkeeper Matt Turner from distance in the 11th minute, and the home side managed to get themselves back in the match in the 17th minute.
Smart work from Rashford set up the goal, with the Englishman driving into the box before delivering into Eriksen, who finished into the back of the net from close range.
Man United were firmly on the front foot and should have levelled in the 27th minute when the ball bounced towards Casemiro from a corner, but the midfielder somehow guided his header wide of the post in one of the misses of the campaign so far.
Forest's main threat was coming on the break, and Steve Cooper's side had a chance to score a third late in the first period when Brennan Johnson pulled a cross back for Gibbs-White, but his rasping effort hit his own teammate Awoniyi, as the visitors entered the half-time break with a 2-1 lead.
The away team had another chance early in the second period, with substitute Victor Lindelof, who replaced Raphael Varane, caught on his heels, but Lisandro Martinez managed to block the shot from Awoniyi.
Fernandes sent one over the Forest crossbar shortly after, but Man United were level in the 52nd minute when Casemiro converted from close range following a knockdown from Fernandes, who had initially played his free kick back to Rashford in a training-ground move.
Forest goalkeeper Turner made a brilliant save to keep out a curler from Antony in the 54th minute, before Fernandes missed the far post from an Antony cross just before the hour.
The visitors were reduced to 10 men in the 67th minute when Worrall was handed a red card for a pull on Fernandes just outside the penalty box.
Man United were awarded a penalty in the 75th minute when Rashford was brought to the ground by Danilo, and Fernandes stepped forward to send the home side 3-2 ahead to complete the turnaround.
Serge Aurier just failed to meet a header from the impressive Awoniyi in the 80th minute, with Forest looking for a third of their own, leading to the introductions of Chris Wood and ex-Red Devils attacker Anthony Elanga.
Man United goalkeeper Andre Onana made a smart save to deny Wood in the 84th minute, before Elanga fired over the crossbar from a narrow angle five minutes later.
Wood had a shout for a penalty when Onana appeared to block the striker from reaching a cross, and there were 11 minutes of additional time at Old Trafford, but the Red Devils managed to hold on to secure a huge three points in their third match of the new season.
Man United will return to league action away to Arsenal next Sunday afternoon, while Forest will take on Burnley in the second round of the EFL Cup on Wednesday night.