Matt Andriese, Adam Ottavino and Matt Barnes each worked a scoreless inning in relief. Barnes earned his third save.
"I was really excited as soon as I went out there," Rodriguez said. "It feels amazing to be back here to step on the mound and be able to compete. ... That was something special for me."
Rodriguez was pulled after allowing his second homer of the day, a towering leadoff shot by Randal Grichuk leading off the seventh. Bo Bichette also homered for the Blue Jays, who have lost three straight.
Manager Alex Cora said he likes the way has Rodriguez attacked the strike zone and set up the bullpen this season.