Brady completed 22-of-36 passes for 284 yards with two scores, moving past Johnny Unitas for the second-longest streak in NFL history as New England (12-4) earned the No. 2 AFC seed and secured a first-round bye.
Drew Brees of New Orleans holds the record for consecutive games with at least one touchdown pass at 54 -- a streak that was snapped in October.
Tight end Rob Gronkowski returned from a five-game absence to catch a touchdown pass and Justin Francis led the Patriots' defensive charge with three sacks in the victory.
Miami (7-9) fell as rookie quarterback Ryan Tannehill was limited to a 20-of-35 passing effort for 235 yards with an interception and a lost fumble.