A determined Lionel Messi led the Spanish champions' response as the Argentina forward created Pedro Rodriguez's 36th-minute equalizer before David Villa curled a long-range free kick into the net five minutes into the second half.
"We could have played better after Barcelona took the lead," AC Milan coach Massimiliano Allegri said. "But I'm happy because the team did not collapse until we found the late equalizer."
That came deep into stoppage time when Thiago leapt high to meet Clarence Seedorf's corner to secure a point for Milan.
"We did all we were supposed to do but we are still left with this bitter taste," Barcelona coach Guardiola said.
All four teams in Group H are on one point after BATE Borisov and Viktoria Plzen drew 1-1.
In Germany, Arsenal went in front just before halftime when Van Persie won possession from Sebastian Kehl and played a one-two with Theo Walcott before beating Dortmund goalkeeper Roman Weidenfeller with his shot.
But after Marcel Schmelzer's free kick was only half-cleared by Kieran Gibbs toward the end, Ivan Perisic volleyed in a late equalizer.
The draw enabled Marseille to go top of the group after Lucho Gonzalez secured a 1-0 victory over Olympiakos.
The Argentine exploited a lapse in the Greek defense, meeting Jeremy Morel's cross before beating former River Plate teammate Franco Costanzo in the Olympiakos goal.
AP