Quarterback Matt Ryan completed 8 of 11 passes for 69 yards in the first quarter, but the 13-play, 75-yard drive stalled at the 6-yard line when he overthrew Julio Jones and Tony Gonzalez in the end zone.
Brady was 2 of 4 for 39 yards as the Patriots relied mostly on running back Stevan Ridley, who ran five times for 35 yards.
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.
The Falcons entered the game 1-2.
Brady had moved New England to the Atlanta 6-yard line when the first quarter ended.
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.
AP NFL website: www.pro32.ap.org
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