It looked as though Jose Mourinho's side would be heading for a second defeat at the hands of the Italian champions in the space of two weeks after Cristiano Ronaldo had opened the scoring in stunning fashion shortly after the hour mark - a result which would have left United third in Group H following Valencia's win over Young Boys earlier in the day.
Juventus hit the woodwork twice and squandered a host of golden chances to add to their advantage in the second half, and they were punished for that profligacy in the closing stages as United produced a late show to inflict a first defeat of the season on their hosts and give an unexpected boost to their own qualification hopes.
Juan Mata levelled things up with an 86th-minute free kick before Alex Sandro inadvertently turned the ball into his own net four minutes later to secure a 2-1 victory for the Red Devils which will rank among their memorable Turin wins in the past, such as the 3-2 triumph in the 1999 semi-finals and a 3-0 victory on their last visit in 2003.
United were more competitive from the off than they were during the whole of the reverse fixture at Old Trafford as they put Juventus under increased pressure on the ball, although the visitors still lacked a cutting edge in the final third until the closing stages.
Almost all of the goalmouth action fell to Juventus during the opening 45 minutes, with Rodrigo Bentancur seeing one effort deflected wide before Ronaldo fired his first shot of the night past David de Gea's post.
One of United's best chances to test Wojciech Szczesny in the first half came from a free kick just outside the area, but Alexis Sanchez's shot hit the wall and within seconds Juve were up the other end as Sami Khedira forced a save from De Gea with a powerful shot.
The home side's pressure did begin to build shortly after the half-hour mark and De Gea was forced into his first testing save of the evening when he reacted sharply to keep out a Juan Cuadrado cross which deflected goalwards off the legs of Nemanja Matic.
De Gea was helpless just a minute later when Ronaldo cut the ball back for an unmarked Khedira inside the area, but the German - drafted into the starting lineup in place of Blaise Matuidi - dragged his finish against the post when he should have scored.
Despite United holding their own for the most part, it took until first-half stoppage time for their first shot on target to arrive, and even then it was a simple one for Szczesny to gather from Pogba's speculative long-range effort.
The visitors did create a much better chance in the opening stages of the second half, though, with the in-form Anthony Martial cutting inside from the left channel and curling a finish narrowly past the post when looking for the bottom far corner.
United were rescued by the woodwork for a second time just a minute later as Juventus responded with a stunning Paulo Dybala effort which turned De Gea into a mere spectator but crashed against the crossbar.
If De Gea was left dumbfounded and helpless by the quality of that effort, the opening goal was past him before he could even react as Ronaldo ended a five-game European drought and scored his first Champions League goal for Juventus in style.
Leonardo Bonucci's pass over the defence was exquisite, and Ronaldo watched it all the way over his shoulder before connecting with the sweetest of volleys to fire a goal of the season contender past his former club.
The floodgates appeared to have opened after the deadlock was broken and Ronaldo almost turned provider moments later when he teed up Miralem Pjanic, who saw his first-time curling effort palmed away by De Gea.
Pjanic should have done better with another chance in the 73rd minute before Juventus were on the attack again one minute later, only for Cuadrado to spoon a glorious opportunity over the crossbar after exchanging passes with Ronaldo.
Even accounting for Juve's wastefulness, there only appeared to be one winner at that stage and De Gea was left rooted to the spot for a fourth time in the match when he watched another swerving Pjanic drive fly narrowly over the crossbar.
It was an earlier change - Marcus Rashford - who offered the first glimpse of a fightback when he turned sharply and sent a long-range shot bobbling wide, but Mata was more accurate from outside the area with four minutes remaining.
Martial threatened twice in quick succession to contribute to a free kick in a good position, and Mata stepped up to curl his finish over the wall and beyond the despairing dive of Szczesny - the first goal Juve had conceded in this season's competition.
A point would have been enough to secure Juventus' place in the last 16 with two games to spare, but the hosts were in no mood to settle and squandered another sight of goal shortly after the equaliser when Dybala drilled well wide from the edge of the box.
Massimiliano Allegri's side still may not have envisioned being further punished, though, and the winner arrived in the final minute of normal time when Ashley Young's inswinging free kick from deep caused enough problems that Szczesny could only parry it out against Bonucci, who was helpless to prevent it from then bouncing off Sandro and creeping over the line.
Bonucci tried to atone for the own goal with a long-range drive moments after the restart, but it was United who had the biggest chance to further add to the scores when Martial released Rashford clean through on goal, only for Szczesny to deny the England international.
It did not prove to be a costly miss, though, as Juve had no time to find a response en route to only their second defeat in 36 home Champions League group games since United's last visit to Turin in 2003.
Indeed, it is just the third time Juventus have lost in Europe at the Allianz Stadium, while United are now the only team to have won three away Champions League games against the Bianconeri - doing so in successive matches in 1999, 2003 and 2018.
The result leaves the Red Devils second in Champions League Group H, now only two points adrift of Juventus and two above third-placed Valencia with qualification still firmly in their own fans.
United will now look to harness the momentum from this victory - which comes just a fortnight after being outclassed by the same opposition at Old Trafford - when they take on free-scoring Premier League champions Manchester City in the derby at the Etihad Stadium on Sunday.
JUVENTUS (4-4-2): Szczesny; De Sciglio (Barzagli 83'), Bonucci, Chiellini, Sandro; Cuadrado (Mandzukic 92'), Khedira (Matuidi 61'), Pjanic, Bentancur; Dybala, Ronaldo
MAN UTD (4-3-3): De Gea; Young, Smalling, Lindelof, Shaw; Herrera (Mata 79'), Matic, Pogba; Lingard (Rashford 70'), Sanchez (Fellaini 79'), Martial