After Toronto's six-run fourth inning, which included a grand slam and two fielding gaffes, Cleveland trailed 6-1 and appeared en eroute to another defeat.
Francona and several players spoke up in the dugout, and the Indians rallied.
Pinch-hitter Ryan Raburn's tiebreaking, two-run double capped a six-run fifth inning, Jason Kipnis had four hits and the Indians defeated the Blue Jays 10-7.
"Guys were very forward in being vocal and said what needed to be said," Kipnis said. "We're not good enough to play that stupid the way we did in the first couple of innings."
Kipnis also homered and drove in two runs for the Indians, who overcame Devon Travis' first career grand slam.
"That fourth inning was a tough one to take on a lot of levels," Francona said. "Whether we hit a switch or whatever, we played the game the way you're supposed to. We got after it and stayed after it."
Michael Brantley's two-run single off Jeff Francis (1-1) tied the score in the fifth and Raburn, batting for David Murphy, lined a double over third for an 8-6 lead.
Marc Rzepczynski (1-0) pitched a scoreless inning for the Indians, who gained a four-game series split with just their fourth win in 12 home games this year.
Raburn, who hit .200 with 22 RBIs during an injury-filled 2014, is batting .400 with 10 RBIs in 40 at-bats, including a .457 average (16 for 35) against left-handers.
Cleveland starter Trevor Bauer allowed six runs, seven hits and three walks in 4 1-3 innings. Toronto's Drew Hutchison gave up six runs and eight hits in 4 1-3 innings.
Mental mistakes in the field by Bauer and first baseman Carlos Santana helped Toronto to its early lead.
Kipnis hit a solo homer in the third, but the Blue Jays put up their second six-run inning in as many days.
Santana ran across the infield to catch Kevin Pillar off third base after fielding Ryan Goins' second-inning grounder but failed to make the tag, allowing Toronto to load the bases.
Francona said Santana began the play correctly, but should have thrown the ball to Lonnie Chisenhall as he neared third base.
"Up until he didn't give the ball up, he did it textbook," Francona said. "So then the textbook went away."
Bauer fielded Ezequiel Carrera's high chopper but didn't get an out as Pillar scored for a 2-1 lead.
Toronto went 3-7 on its longest scheduled trip this season. Russell Martin hit a solo homer off Cody Allen in the ninth.
Bauer admitted he was "extremely frustrated" after the fourth inning but was happy with the final results.
"For one of the few times this year we had guys step up and lead," he said. "That's a big thing for a team. The leadership has been kind of a question of who's going to take that role and where it's going to come from. Guys performed when we needed them to and guys said certain things. When you look at the leadership picture it was a good day for us. "
Blue Jays: SS Jose Reyes (left rib fracture) has been on the 15-day DL since April 28.
Indians: CF Michael Bourn returned to the lineup after missing a game because of a strained neck. He landed face-first sliding into the plate Friday. Bourn remained in that game but was a late scratch Saturday.
Blue Jays: RHP R.A. Dickey, who is 0-3 in five starts, faces the visiting New York Yankees in the opener of a three-game series Monday. Dickey held New York to one run in 6 1-3 innings on April 8.
Indians: Cleveland opens a three-game series Tuesday at Kansas City. RHP Danny Salazar, who defeated the Royals on April 29 will start the opener.