The 2-1 victory is United's latest memorable win in Turin and boosts their hopes of progressing from Group H to the knockout rounds.
Jose Mourinho's side looked on course for back-to-back defeats against the Italian champions following their 1-0 home defeat two weeks ago.
He told the channel: "As soon as there was the free kick in there, I had the confidence that I had to take it".
Back in 1999, United's 3-2 semi-final comeback against the same opposition reminded the world never to write them off on their way to the Nou Camp.
Like most foes who go to Juventus, United had been outplayed, but they still played well enough to stay in the game as Rodrigo Betancur, a 21-year-old midfielder who cost only €9 million (Dh37.75m), impressed as much as he had at Old Trafford.
This time he had another win to celebrate, after watching his team fight back from Cristiano Ronaldo's spectacular opening goal.
"I probably shouldn't have done it", he admitted, via The Times.
The result leaves United in a much more comfortable position in the group stages, now having only to beat Young Boys and draw with Valencia to qualify for the last-16. "We should have won the game easily, by three or four goals".
"But the team did well, we are still at the top of the group and we have two matches which will allow us to finish first".
Allegri was asked if he would have reacted similarly to Mourinho.
"The love that they showed me today was unbelievable and how great is football when you see that?"
DF Victor Lindelof, 6 - fearless block from a Ronaldo shot early on but lost him when Leonardo Bonucci floated a ball over the top to allow the former United man to smash Juventus into the lead.
It's unclear exactly what was said, but it's clear he was provoked. They bought that player (Ronaldo), they want to win everything, they can win everything and we are a team where numerous players didn't play big matches like this in the Champions League.
Sandro's night may have ended in misery but the Brazilian had a moment to remember as he got past Lingard by putting the ball through his legs on the wing during the first half.