Blackley (3-2) struck out eight, allowing only one run on five hits while issuing a single walk for the A's, who notched a season-high sixth straight victory.
The 29-year-old Australian left-hander tossed 65 of his 97 pitches for strikes, as the surging Athletics improved to 15-2 in July, putting themselves in contention for the American League wild-card post-season berth.
Clinging to a 2-1 lead in the seventh, the A's exploded for five runs, highlighted by a bases-clearing single from Cespedes to put the contest away.
Derek Norris cracked a two-run homer in the win.
Toronto starter Brett Cecil (2-3) absorbed the loss despite allowing only two runs on five hits over six frames.
J.P. Arencibia and Travis Snider each cracked a homer for the Blue Jays.