Cristiano Ronaldo looked to have haunted the club which turned him into a global superstar when he volleyed in the opening goal in the 65th minute after Juventus had dominated proceedings and struck the woodwork twice.
Jose Mourinho's side had looked toothless in attack but after the Portuguese threw on Mata, Marouane Fellaini and Marcus Rashford late in the game they staged the most unlikely of comebacks to revive memories of their epic 3-2 win from two goals down in Turin in the 1999 semi-finals.
Spaniard Mata equalised with a perfectly-executed free kick in the 86th minute and United caused bedlam in the Juve area with another set-piece, with the ball appearing to go into the net off Alex Sandro after a knockdown from Fellaini.
Competition organisers UEFA later credited Juve defender Leonardo Bonucci with the goal, which meant United became the first English team to beat Juventus away from home in 15 years, when a previous United team, inspired by Ryan Giggs, won 3-0 at the old Stadio dello Alpi in 2003.
The stunning finale prevented Juventus from sealing their place in the last 16. They still lead Group H with nine points but United are second on seven, two ahead of Valencia in third place, while Young Boys are bottom of the group and can no longer finish in the top two.
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Juventus' Cristiano Ronaldo reacts after Manchester United's second goal. (Reuters)
Defeat would have likely obliged United to beat Young Boys at home and Valencia away to qualify. Now they can seal qualification for the last-16 with a victory against the Swiss side if Valencia fail to beat Juventus in their next fixture.
Juve's dominant performance at Old Trafford two weeks ago appeared to underline the current gulf in class between the two European greats, which was also reflected in their respective domestic results: Juve are riding high, six points clear at the top of Serie A, while United are seventh in the Premier League.
Juve began strongly and almost went ahead when Sami Khedira side-footed a cross from Ronaldo against the far post.
The Italian champions again struck the woodwork in the second half when Paulo Dybala clipped the crossbar from outside the area, and there was more than an air of inevitability about Ronaldo's deadly volley, created by a classy chipped pass from Bonucci, which broke the deadlock.
This time there was to be no muted celebration from the Portuguese who lifted his shirt to flex his chest muscles before high-fiving team mates.
He could hardly have envisaged the stirring comeback his old side would later conjure, which ended with Mourinho being booed off the pitch after appearing to taunt the home fans.
Results of other matches:
Real Madrid 5-0 Viktoria Plzen
Manchester City 6-0 Shakhtar Donetsk
Bayern Munich 2-0 AEK Athens
AS Roma 2-1 CSKA Moscow
Valencia 3-1 Young Boys
