Brady, who had a streak of 52 straight games with a touchdown pass snapped last week, engineered the game-winning drive with 73 seconds left in Sunday's contest in front of a crowd of 68,756 at Gillette Stadium.
He moved the ball from his 30 to the Saints' 26-yard line with three completions.
Facing fourth down and four, Brady hit Austin Collie for a nine-yard gain that kept the game-winning drive going.
After spiking the ball to stop the clock, Brady hit Thompkins in the back left corner of the end zone for the win.
"Our defence made a great stop," Brady said. "The guys made a bunch of plays there at the end. Guys made big-time catches."
Brady totalled 269 yards on 25-of-43 passing for New England, who improved to 5-1.
Bill Belichick collected his 210th win as a head coach, passing Chuck Noll for sole possession of fifth place on the NFL's all-time list .
Jamaal Charles had two short touchdown runs and the Kansas City Chiefs sacked quarterback Terrelle Pryor 10 times in a 24-7 win over the Oakland Raiders.
The Chiefs improved to 6-0 and snapped a six-game home losing streak to the Raiders.
This is the first time since 2003 they have won the first six games of the season.
In other games, Knowshon Moreno rushed for three second-half touchdowns and the Denver Broncos remained unbeaten with a 35-19 win over the Jacksonville Jaguars.
Quarterback Peyton Manning added two more touchdown passes to his season total and finished with 295 yards on a 28-of-42 passing to help Denver to its 17th consecutive regular-season victory.
In East Rutherford, New Jersey, the Pittsburgh Steelers finally broke into the win column with a 19-6 victory over the New York Jets.
Detroit quarterback Matthew Stafford had four touchdown passes, three of them to Joseph Fauria, as the Lions used a strong second half for a 31-17 win over the Cleveland Browns.
In other games it was Carolina 35 Minnesota 10, St Louis 38 Houston 13, Green Bay 19 Baltimore 17, Philadelphia 31 Tampa Bay 20, Cincinnati 27 Buffalo 24, Seattle 20 Tennessee 13, San Francisco 32 Arizona 20, and Dallas 31 Washington 16.