The north Londoners looked destined to drop into the Europa League after Ousmane Dembele put Barca into an early lead , while Maurco Icardi had pulled Inter level with PSV after falling behind to a Hirving Lozano goal.
But Lucas Moura struck from Harry Kane's cross late on to secure the draw, though Tottenham continued to attack unaware that the point would be enough to take them through.
And captain Hugo Lloris told BT Sport: "It is true that until the end we had to play for the three points and for the win and try to not concede a goal as well.
"It was difficult but the spirit was fantastic.
"I understood we had gone through when I heard the fans, they made such a noise so it was fantastic."
Striker Kane added: "We weren't sure what the [Inter-PSV] result was.
"We deserved at least a draw in this game and we're just buzzing that we're through.
"We had to go for the win and thankfully PSV got the draw for us. A draw away to Barcelona is a great result."
Tottenham's fans were in jubilant mood at full-time, though their celebrations appeared to take place some two minutes before the full-time whistle at San Siro.
Nevertheless, the visitors produced a display worthy of a place in the knockout phase and Lloris said their spirit helped them recover from going behind early on.
World Cup winner Lloris added: "It was difficult for us after the early goal to come back in the game.
"But we stuck to our principals and to the basics and we produced a good performance.
"We didn't win but we're though and that is the most important thing."