Callaspo had committed an error that allowed Toronto tie the score 1-1 in the eighth but came through with his second hit of the day the next inning.
Casey Janssen (4-1) came on to pitch for the Blue Jays in the ninth and Josh Donaldson hit his second pitch for a single. After an out, Brandon Moss delivered pinch-hit double and Josh Reddick was intentionally walked, loading the bases,
Callaspo then pulled a hit down the right-field line for a double that scored Donaldson and Moss. Stephen Vogt followed with an RBI single and Eric Sogard added a sacrifice fly for the 5-1 Oakland lead.
That gave the win to Ryan Clark (5-2), who got the final two outs of the eighth inning.
Both starting pitchers were sharp. Toronto's J.A. Happ allowed three hits and one run -- a lead-off home run by Chris Young in the first -- while striking out six and walking two. Dan Straily gave up six hits and one run in 7 1/3 innings for Oakland. He struck out five and walked one.
Straily had a four-hitter through seven but allowed two singles in the eighth and was lifted. The first batter reliever Ryan Cook faced reached on an error that allowed the tying run to score.