Brady's two scoring strikes came just 30 seconds apart as the AFC East-leading Patriots (10-3) come from behind for a victory for the third straight game.
The triumph, however, was nearly overshadowed by the loss of all-pro tight end Rob Gronkowski, who was taken off the field in the third quarter after suffering a suspected torn knee ligament.
Gronkowski was hurt after making a reception and taking a hit from the Browns' T.J. Ward.
The team said he is set for magnetic resonance imaging exam Monday.
Quarterback Jason Campbell returned from a concussion and played well for Cleveland (4-9). His third TD pass of the game handed the Browns a 26-14 lead with 2:39 left.