Mauricio Pochettino’s side were 2-0 down inside 10 minutes after Gonzalo Higuain followed up a sumptuous volley with a penalty, awarded for Ben Davies’ foul on Federico Bernardeschi.
Harry Kane started the fightback after rounding Gianluigi Buffon following Deli Alli’s threaded pass and Higuain then missed a penalty for the hosts with the last kick of the first half, following Serge Aurier’s poor challenge on Douglas Costa.
Spurs dominated possession against the Italian champions and got their reward when Christian Eriksen’s low free-kick evaded Buffon to give Spurs the edge going into the second leg at Wembley on 7 March.
Spurs remain unbeaten since 16 December, and Pochettino said: “We are more mature. The team is growing and have stepped up in this aspect. I think the team was amazing in the way they reacted.”