The A's had grabbed a 2-0 lead in the third inning on RBI from Josh Donaldson and Josh Reddick following Asdrubal Cabrera's throwing error before Cespedes capped the uprising with his two-run shot.
The Oakland left fielder later added a run-scoring single, while Crisp's three-bagger gave the A's an 8-1 lead in the eighth.
Bartolo Colon (10-9) earned the victory, holding the Indians to one run on five hits through eight innings. The veteran right-hander didn't allow a walk and struck out three.
Grant Balfour was called on to make his 11th save after the A's bullpen coughed up four runs in the ninth.
Oakland, which came into the contest a half-game out of an American League wild card spot, clinched the three-game series with the triumph.
The Athletics got to Indians starter Corey Kluber (0-2) for four unearned runs on four hits over five full innings.