The Marlins had forged a 2-2 tie with two runs in the sixth, but the Indians quickly responded in the next half-inning.
Kipnis (3-for-4, three RBI) followed by lining a pitch up the middle, easily scoring Stubbs. Bourn followed him to the plate, beating the throw home with a slide in a controversial call.
The extra run turned out to be key when Miami's Greg Dobbs drove in a run in the ninth against Tribe closer Chris Perez, who recorded the game's final two outs with runners at second and third for his 16th save.
Cody Allen (5-1) notched the decision as second-place Cleveland rebounded from a 10-0 shellacking in Friday's series opener to remain three games behind American League Central-leading Detroit.
Christian Yelich and Ed Lucas each provided an RBI for the Marlins.