Bichette, who had been out with a right quad injury, had an RBI double in to tie the game at 1-1 in the sixth and an RBI single in the Jays' four-run seventh.
Dairon Blanco hit a solo home run for the Royals (44-98) as the teams opened a three-game series.
Royals right-hander Collin Snider allowed a hit and a walk in two scoreless innings. Right-hander Alec Marsh replaced him and allowed one run, three hits and two walks with six strikeouts in four innings.
Left-hander Yusei Kikuchi allowed one run, three hits and one walk while striking out eight in five innings for Toronto (78-63).
Blanco hit his second home run of the season with two out in the third.
Kevin Kiermaier led off the Toronto sixth with a triple and scored on Bichette's double. Davis Schneider, who doubled twice in his first two at-bats, walked later in the inning, and Cavan Biggio walked to load the bases. Marsh struck out former Royals player Whit Merrifield to end the threat.
The Royals regained the lead in the seventh against Trevor Richards. Freddy Fermin blooped a two-out single to center and scored on a triple to right by Nick Loftin. Kyle Isbel walked. Chad Green (2-0) replaced Richards and struck out Blanco to end the inning.
Austin Cox retired one batter in the home seventh before leaving the game after he suffered a leg injury covering first base on Alejandro Kirk's infield hit. Carlos Hernandez (1-10) took over, and got the second out before walking George Springer. Bichette then singled in one run, and Vladimir Guerrero Jr. hit a two-run double for a 4-2 lead. After a walk, left-hander Tucker Davidson entered and allowed an RBI single to pinch hitter Ernie Clement.
In the top of the eighth, Green allowed a double to Bobby Witt Jr. and a single to Salvador Perez before giving way to Tim Mayza. A run scored on pinch hitter Edward Olivares' fielder's choice grounder. After a single and a walk loaded the bases, Toronto closer Jordan Romano entered and got an inning-ending grounder.
Romano worked around two walks and Perez's RBI single in the ninth to get his 33rd save.
—Field Level Media