Trailing 30-21 in the fourth quarter, Baltimore (2-1) rallied for the game's final 10 points to stun the Patriots.
Torrey Smith -- who played the game less than 24 hours after the death of his younger brother in a motorcycle accident -- caught his second touchdown pass with 4:08 remaining, cutting the Ravens' deficit to two points.
Joe Flacco (28-of-39, 382 yards, three touchdowns) then drove Baltimore 70 yards on seven plays with the clock winding down, capped by Tucker's game-winning boot.
Smith finished with six grabs for 127 yards, while Ray Rice amassed 101 rushing yards and a TD on 20 carries for the Ravens.
Tom Brady completed 28-of-41 passes for 335 yards and a score for New England (1-2), which fell under .500 for the first time since 2003.