Trailing 5-2 entering the ninth and facing Indians closer Chris Perez (2-1), the Red Sox launched a rally when Dustin Pedroia drew a leadoff walk, went to second on a David Ortiz double and then scored on Mike Napoli's fielder's-choice ground ball.
Ortiz then stole third and came home on Jarrod Saltalamacchia's groundout, making it 5-4 with two outs and the bases empty.
Joe Smith came in to pitch and Ellsbury delivered a line-drive double to center field, sending home Gomes and Drew (3-for-4, three runs scored) for the game-winning margin.
Craig Breslow (2-0) earned the decision despite yielding a run in the ninth as Boston took three-of-four games from Cleveland and claimed a share of the American League East lead.
Nick Swisher knocked a solo homer and added a sacrifice fly while Jason Kipnis also went deep for the Indians.