Bowers, a redshirt sophomore who had never thrown a collegiate pass until this season opener, completed 24 of 38 passes for 363 yards.
The go-ahead points came when receiver Jordan Duncan caught a 20-yard touchdown pass with 12:58 remaining. That was one snap after the Bears converted a fourth-down play when 285-pound fullback Malik McMorris caught a pass.
California running back Vic Enwere added a 1-yard touchdown run with 1:56 to play before Tar Heels quarterback Chazz Surratt scored on a 1-yard run on the game's last play.
Freshman running back Michael Carter scored on two touchdown runs for the Tar Heels in the first-ever meeting between the teams.
Surratt's 4-yard touchdown pass to Thomas Jackson with 2:02 remaining in the third quarter sent the Tar Heels to a 24-21 lead. That came two plays after linebacker Andre Smith's 73-yard interception return.
Earlier in the quarter, Patrick Laird turned a short pass into a 54-yard touchdown to give the Bears a 21-17 lead.
Carter gained 71 first-half yards on the ground. He ended up with 94 yards on 11 carries.
The Bears scored first on receiver Jordan Veasy's 10-yard reception from Bowers.
Carter answered with a 2-yard touchdown run for North Carolina.
The Tar Heels led for the first time with 8:21 remaining in the second quarter on Freeman Jones' 39-yard field goal on the junior's first career field goal attempt.
California closed within 17-14 before halftime on receiver Vic Wharton's 67-yard catch-and-run.