The Falcons went three-and-out on their second possession due in part to a holding penalty on left tackle Sam Baker that wiped out an 18-yard run by Jacquizz Rodgers.
Atlanta, which began the game with a 1-2 record, failed to convert a fourth-and-2 from the New England 7 midway through the second when Ryan threw behind Roddy White near the left sideline.
New England, off to its best start in six years, was trying to improve to 4-0 and tie Denver and Kansas City for the AFC's best record.
The Patriots had yet to face a quarterback this season as successful as Ryan. Others were rookies EJ Manuel of Buffalo and Geno Smith of the New York Jets and Josh Freeman of winless Tampa Bay.
New England's defense had allowed just three touchdowns this year, but the Patriots' offense has only four touchdowns in 13 trips inside the opponents' 20-yard line for an NFL-worst 69.2 percent through three games.
Atlanta entered the game with a 34-7 record under coach Mike Smith at the Georgia Dome, but his defense looked confused on a 49-yard pass from Brady to Thompkins that set up Gostkowski's field goal.
Brady faked to Julian Edelman on a reverse and connected with Thompkins, who jumped to beat strong safety William Moore for the big gain.
Brady attempted just nine passes in the first half, completing five for 96 yards, but he has thrown a TD pass in 52 consecutive games, the NFL's second-longest career — and active — streak behind Drew Brees' 54 straight for New Orleans.
AP NFL website: www.pro32.ap.org
16 Week Curriculum With Instructions, Lesson Plans & CNG Conversion Kit