New England improved to 4-0 for the first time since its near-perfect season in 2007.
"We slugged it out," Brady said. "We came to Atlanta, which is a tough place to play and they've got a very good team. It's nice to win and be 4-0."
"We've got to do a better job of finishing out the game," New England coach Bill Belichick said, "but we did a good job of putting ourselves in that position with a 17-point lead with six minutes to go in the game."
Thompkins, an undrafted free agent, finished with six catches for 127 yards -- nearly as much as the first three games of his career.
"He's an impressive young player," Brady said. "It's really rare to see that with a young player, to step in with confidence like that he has."
"We're missing quite a few guys," Brady said, "so hopefully when we get them back, we'll be more efficient."
Atlanta (1-3) dropped two games below .500 for the first time since 2007 despite a career night from Tony Gonzalez. He had 12 catches for 149 yards and a pair of touchdowns.
"You get what you earn in the NFL," coach Mike Smith said. "We've earned a 1-3 record through the first quarter of the season. That's what we are."
New England led just 13-10 before a wild final period. Blount stunned the Georgia Dome crowd with his burst through a gaping hole, cutting outside and dragging defenders into the end zone.
Edelman hauled in a 16-yard pass, followed by 44-yard reception that set up the touchdown throw to Thompkins in the left corner of the end zone, sending many of the Atlanta fans heading to the exits with nearly 10 minutes left.
Gonzalez hauled in an 11-yard touchdown pass, the Falcons recovered an onside kick, then drove for another field goal by Bryant.
New England recovered a second onside kick, only to fumble a snap trying to convert on fourth down, turning it back over to the Falcons.
Atlanta scored only one touchdown on six trips inside the 20. That included settling for a field goal on their first possession, when they had first-and-goal at the 6, and Smith's decision to go for it on fourth-and-2 from the 7. That failed when Ryan threw it behind White in the flats.
"We're just not executing how we need to in the red zone," Smith said. "We're moving the ball well between the 20s. But we're not ending with touchdowns. You've got to score touchdowns in this league. It's an offensive league."
New England got good production from its running game, especially on its first touchdown drive. Brady threw only twice on the 12-play, 75-yard possession, including a 1-yard touchdown pass to third-string tight end Matthew Mulligan.
NOTES: Ryan completed 34 of 54 for 421 yards. ... The Falcons had won 14 in a row after losses. This is their first two-game losing streak since 2009. ... Atlanta, which faces the New York Jets in Week 5, has never lost three in a row under Smith. ... The Patriots rushed for 132 yards. Blount led with 64 yards on nine carries, while Ridley chipped in with 53 yards on 11 carries. ... In addition to breaking up Atlanta's final pass, Talib also had an interception in the fourth quarter - his fourth of the season.
Copyright 2013 by The Associated Press