The record was hit with a 15-yard completion to running back Leonard Fournette during the fourth quarter which was followed up by a field goal from Ryan Succop.
Brady ended the 16-13 win with 280 yards through the air and one touchdown, while Rams quarterback Matthew Stafford sat on 165 yards and also managed a touchdown.
The Buccaneers opened the scoring with a field goal but the Rams led for most of the game after Cooper Kupp found a touchdown from a 69-yard pass from Matthew Stafford in the first quarter.
The visitors extended their lead with field goals despite the Buccaneers kicking two more of their own.
But it was the star of the day, Brady, who led the home side back to victory with a pass to Cade Otton who caught the touchdown late in the fourth quarter.