Clint Dempsey swept the ball past David de Gea with seconds remaining to cancel out Robin van Persie's first-half opener at White Hart Lane.
And Villas-Boas believes a share of the points was the least Spurs warranted after dominating much of the game.
"It was important for us. We dominated the whole game - possession and shots," Villas-Boas told Sky Sports.
"We controlled the game and played well. We deserved to win, to be fair.
"But Manchester United made the most of their opportunity and we were chasing against them, which is very difficult.
"But 60 per cent possession against United and 15 shots against four - we should have gone out with a win. We deserved this last-minute goal.
"Obviously we set our targets high. We want to win. We understand against a team like United you are not going to go two or three goals up.
"But we managed to at least get a point and I think a deserved point.
"It might be the game where United have the least opportunities ever in the Premier League.
"It gives you a boost playing so well and getting the goal in the last minute."
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