It was a 2-2 deadlock heading into the ninth before the teams combined to send 15 men to the plate. Each side scored an unearned run in the ninth.
The Blue Jays loaded the bases with two out in the top half of the inning before Kawasaki singled to center.
In the bottom half of the inning Asdrubal Cabrera delivered a two-out single that scored Bjorn and Cabrera came in later in the inning on a throwing error by Kawasaki at shortstop.
It was the first save of Delabar's three-year career and preserved the victory for Neil Wagner (2-3), who retired the only man he faced in the eighth inning after the Indians had scored a run to tie the game.
Rich Hill (0-1) loaded the bases before being relieved in the ninth and suffered the loss. This is Hill's ninth major league season and he had not been tagged with a loss since 2009, when he was with Baltimore.
Swim Week Miami Beach 2014 [PHOTOS]