Bryant kicked a 49-yard field goal with eight seconds left as the Falcons had squandered a 20-point second-half lead.
"I never cried after a win before," Tony Gonzalez, the Falcons veteran receiver, said afterwards.
"For us to come back the way we did. I am so proud of our guys. We knew they would make a run."
Quarterback Matt Ryan tossed three touchdown passes en route to his first playoff victory for the Falcons, who had lost their last four playoff contests.
Running back Michael Turner carried the ball 14 times for 98 yards and fifth-year Falcons quarterback Ryan completed 24-of-35 passes for 250 yards.
Seattle rookie quarterback Russell Wilson had looked poised to continue his Cinderella run Sunday but his sensational season ended with Bryant's field goal.
Wilson finished 24-of-36 for a playoff team record 385 yards with two touchdown passes and one interception. Wilson also rushed for 60 yards and a touchdown. The defeat snapped Seattle's six-game winning streak.
Marshawn Lynch finished with 46 yards and a touchdown on 16 carries for the Seahawks, who have lost their last four NFL playoff quarter-final games.
Atlanta's defence held Seattle scoreless in the first half while the Falcons offence built a 20-0 lead thanks to a pair of touchdown passes from Ryan and two Bryant field goals.
Wilson engineered Seattle's rally, leading the Seahawks to three scoring drives on their first three possessions of the second half.
Lynch busted through for a two-yard touchdown to put the Seahawks ahead, 28-27, with 31 seconds left in the fourth.
That set the stage for Bryant's game-winning field goal.
The last time the Falcons advanced to the penultimate round was in 2004, when Jim Mora was the head coach and Michael Vick was the quarterback.
Ryan took over as the face of the franchise four years ago after disgraced quarterback Vick went to prison for his role in a dogfighting ring.
But Ryan had come up empty in each of his three playoff appearances since then.
Ryan had a superb regular season in which he threw for a team-record 4,719 yards with 32 touchdown passes.
Pete Carroll, the Seahawks coach, backed his side's performance despite their last minute heart-break.
"At halftime, we talked about how we can't get ahead right away," he said.
"It's going to take a while.
"It was an exquisite job of refocusing to a football game after being down like that."