The loss dropped Chicago out of a first-place tie with Detroit in the American League Central.
Jason Kipnis delivered a two-run single in the fourth that allowed the Indians to overcome a 3-1 deficit, Chicago moved in front again on an RBI double by Alexei Ramirez in the fifth and Carlos Santana tied the contest once more by drawing a bases-loaded walk in the sixth.
Cleveland finally jumped ahead when Choo's grounder scored Lou Marson from third and Vinny Rottino gave the Indians an insurance run in the eighth with a homer.
Reliever Matt Thornton (4-9) took the loss, surrendering a run on a hit in 1 1/3.
Swim Week Miami Beach 2014 [PHOTOS]