Scott Van Slyke hit his seventh home run of the season and Andre Ethier drove in a pair of runs for LA, which dropped two straight after a three-game winning streak.
LA starter Hyun-Jin Ryu was in line for the win after allowing two runs on seven hits with eight strikeouts over seven innings before Brian Wilson surrendered the lead.
Cleveland trailed by one when it used a three-run eighth en route to the series win.
Santana and Murphy each moved up to third and second, respectively, on a throwing error from Matt Kemp on the play.
Lonnie Chisenhall was intentionally walked to load the bases before Aviles slapped a single to right that gave the Indians a 5-3 lead.
Van Slyke cut the deficit in half with a solo shot to left in the home half of the eighth.
Bryan Shaw (3-1) picked up the win for a scoreless seventh and Cody Allen recorded the final four outs to secure his eighth save of the season.
Earlier, Raburn put the Indians on the board with a two-run shot to left in the fourth that scored Yan Gomes.
Ryu then helped his own cause in a three-run fifth by hitting his second double of the season that scored Miguel Rojas with two outs.
Dee Gordon and A.J. Ellis each followed with walks before Ethier singled in a pair to give the Dodgers a 3-2 lead.