Brady completed 20-of-31 passes with a pair of TDs, including an 18-yard scoring strike to Thompkins with 9:50 left in the fourth quarter for a 27-13 Patriots lead.
Aqib Talib then intercepted Atlanta's Matt Ryan and Brady led New England 28 yards on six plays to set up Stephen Gostkowski's third field goal of the game -- a 49-yarder for a 17-point advantage with 6:23 left.
The Falcons responded with 10 quick points and were at the New England 10-yard line with a chance for a tying score in the closing seconds but fell short when Talib tipped away a potential TD toss to Roddy White in the end zone.
Thompkins finished with six grabs for 127 yards and Julian Edelman contributed seven catches for 118 yards for the Patriots (4-0).
LeGerritt Blount reeled off a 48-yard touchdown run in the win.
Besides the interception, Ryan finished 34-of-54 for 421 passing yards, two touchdowns.
Tony Gonzalez hauled down 12 passes for 149 yards with two scores and Julio Jones added six grabs for 108 yards for Atlanta (1-3).