Atlanta starter Kris Medlen (1-6) saw his winless streak reach nine straight starts and had to leave the game after just two innings when he was hit on the left calf by a line drive off the bat of Emilio Bonifacio.
Rogers was given his first start in two seasons to help out Toronto's injury-plagued rotation. He went 3 1/3 innings before giving way to eventual winner Juan Perez (1-0), who retired all eight men he faced in his Toronto debut.
Casey Janssen worked the ninth for his 11th save.
Toronto scored all its runs in the opening two innings. Cabrera came home in the first on a fielder's choice grounder by Edwin Encarnacion and then singled home two runs in the second.