In the seventh minute, Milan captain Theo Hernandez pushed forward and fired a hard shot that Tottenham goalkeeper Fraser Forster (Hugo Lloris was injured) saved.
On the return, Brahim Diaz came running and pushed the ball towards the goal with a direct shot.
He then lunged forward and headed the ball into the goal when it looked like it had already crossed the goal line for 1-0.
After 70 minutes of play, Dejan Kulusevski rushed out to replace with quick steps.
Brazil's World Cup player Richarlison took over to give Tottenham a lead but regardless of the manpower, the London team created very few dangers.
A quarter of an hour from the end, Milan had a double chance to extend the lead.
First, substitute Charles de Ketelaere was close to nodding in 2-0, replaced Diaz led from the bench.
A minute or so later, the second substitute, Junior Messias, was even closer to extending Milan's lead with his header.
The return in three weeks
The return in London takes place on March 8.
Zlatan Ibrahimovic has not played a game for Milan this season after a lengthy injury rehab but was on the bench in the league meeting with Torino on Friday.
He is not registered in the Champions League squad.
Tottenham have not scored in the first half in eight of their last nine Champions League games (including tonight).
Tonight, Harry Kane had a ball over the bar in the first half but it was clearly offside in the build-up to the attack.
Before the game, a minute's silence was held for the victims of the earthquake in Turkey and Syria just over a week ago, which claimed 35,000 lives, a number that is rising.