The Patriots again played without star tight end Rob Gronkowski and receiver Danny Amendola, both left at home with injuries.
But the Brady-led offense finally looked like itself after struggling a bit the first three games. New England seemed in control when LeGarrette Blount scored on a 47-yard run, and Brady threw an 18-yard touchdown pass to rookie Kenbrell Thompkins.
Thompkins, an undrafted free agent, finished with six catches for 127 yards — nearly as much as the first three games of his career combined.
"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."
Julian Edelman chipped in with 118 yards on seven receptions.
"We're missing quite a few guys," Brady said, "so hopefully when we get them back, we'll be more efficient."
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.
After Matt Bryant's 45-yard field goal brought the Falcons to 20-13, Brady guided the Patriots on an 80-yard drive that took only five plays and barely 2 minutes.
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.
They nearly missed an epic comeback after Stephen Gostkowski's third field goal pushed the Patriots to their 17-point lead.
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.