The home side dominated the opening period but found themselves 1-0 down after Wayne Rooney broke the deadlock on the stroke of half-time.
Ashley Young's second half brace sealed the win which ensured the Reds kept up the pressure on leaders Manchester City and brought to a pleasing end an extremely tough run of fixtures.
"We certainly carried a lot of luck today," the boss told Sky Sports and MUTV afterwards. "It was a great performance from Tottenham in the first half and maybe we got our tactics wrong.
"We improved in the second half and pushed up. In the first period we tried to drop off and bring them onto us a bit but it didn't work.
"At half-time I told the players to push up on top of their back four a bit more and not to allow them to build their play. We did much better at that, but it needed the second goal to give us the confidence in our game. After that we played very well."
The manager pointed to Rooney's headed opener as the turning point in the match, but was quick to salute some impressive displays in United's backline.
"It was a massive performance from our defenders," added Sir Alex. "Tottenham are a very good team - you have to consider that. That was only their second defeat at home since the start of the season.
"Scoring right on half-time was probably our first strike at goal. It was a nice finish from Wayne but a real killer for Tottenham. They were probably sat in the dressing room wondering how they were