The Indians took a 3-0 lead into game four of the best-of-seven American League Championship Series, but were denied a sweep.
Josh Donaldson homered, Edwin Encarnacion drove in two runs and Aaron Sanchez pitched six strong innings for Toronto, who handed Cleveland their first defeat of the post-season.
"We're going out there playing to win every game," said Donaldson, who belted his first homer of the playoffs. "That's what we've gotta do. We believe we have the team to be able to do that.
"It takes one game at a time, one pitch at a time, one at bat at a time, one everything at a time. The fact of the matter is we were able to get it done today."
Toronto host game five on Wednesday, with the Indians again aiming to secure their first World Series appearance since 1997, when they were beaten by the Florida Marlins in a seven-game thriller.
That marked Cleveland's third World Series defeat since they last won it all back in 1948.
The National League Championship Series between the Cubs and Dodgers was knotted at 1-1 going into Tuesday's game three in Los Angeles.
The Cubs give the ball to pitcher Jake Arrieta, who pitched a no-hitter in his last appearance at Dodger Stadium in August of last year.
"We know he's tough, said Dodgers second baseman Chase Utley, who is 0-for-13 against Arrieta. "He's got electric stuff. For every good pitcher, you try to capitalize on their mistakes. They all make mistakes, some more than others, and you have to try to be ready for those mistakes and try to take advantage of those."