Atlanta outfielder Ronald Acuña Jr. left in the seventh inning with an apparent left ankle injury.
Toronto trailed 2-0 and 4-3 before its 10th comeback win this season and third of the series completed a three-game series sweep.
Danny Jansen, hitting .095, led off the ninth with a single off Will Smith (0-3). Marcus Semien singled, and Bichette doubled to the gap in right-center for a 6-4 lead.
Bichette drove in three runs on two hits. Cavan Biggio knocked out Smith with his second double of the game to drive in another run.
Travis Bergen (1-0) pitched a scoreless eighth. Jordan Romano struck out the side in the ninth.
Acuña led off the game with his major league-high 12th homer, then got hurt while trying to beat out a grounder to third base.
Acuña initially was ruled safe by first base umpire Mike Estabrook before a video review overturned the call. Acuña hopped down the right-field line, then collapsed onto the outfield grass. He was able to walk back to the dugout with a slight limp.
Smith gave up four runs and five hits in the ninth while recording only one out.
Dansby Swanson's two-run homer in the sixth had given Atlanta a 4-3 lead. The Blue Jays pulled even in the eighth when Biggio's double drove in Teoscar Hernández, who walked.
Toronto's Ross Stripling struck out nine, one shy of his career high. He allowed two runs and four hits, all in the first of his five innings.
Acuña pulled Stripling's first pitch, an inside fastball, 442 feet down the left-field foul line. The homer landed near the back of the second level.
Charlie Morton allowed three runs, seven hits and two walks in 4 2/3 innings. Morton's exit ended a streak of Braves starters pitching at least six innings in four consecutive games and six of eight.
Blue Jays: RHP Rafael Dolis (right calf) plans to throw a bullpen session on Friday. He was placed on the 10-day injured list after leaving the May 7 game at Houston with tightness in the calf.
Braves: CF Cristian Pache grimaced while running to first base on a groundout in the second and left with right hamstring tightness. 3B Austin Riley, who was given his first break from starting since April 16, entered. Ehire Adrianza, who started at third base, moved to right while Acuña moved to center.
Blue Jays: LHP Steven Matz (5-2, 4.86) starts when Toronto opens a 10-game homestand Friday night against Philadelphia in Dunedin, Florida.
Braves: LHP Drew Smyly (1-2, 6.12) and Atlanta open a three-game series at Milwaukee on Friday night.