Spurs levelled the scores in second-half injury time, with Clint Dempsey slotting the ball past David de Gea.
Robin van Persie had given United the lead early on with a fine header in the first-half.
Although his team left it late, AVB believes that they did enough across the 90 minutes to deserve at least a point:
“It was important for us. We dominated the whole game – possession and shots,” he 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.”
The Portuguese coach was also delighted with the performance of his defence, who kept the league leaders’ attack very quiet:
“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.”
Van Persie opened the scoring after 25 minutes, heading home at the back post from Tom Cleverley’s cross.
Tottenham pressed for long periods of the second-half and found the equaliser with moments to spare. Dempsey poked home from close range from an Aaron Lennon pass deep into injury time.