"You spot them four runs right out of the gate, it's tough to overcome that," Blue Jays manager John Gibbons said.
Soriano connected again with a leadoff blast in the third, a drive that bounced off the top of the wall and went out.
Happ (3-4) allowed five runs and seven hits in 4 2-3 innings. He walked two and struck out five.
Gardner threw out Toronto's Moises Sierra at home plate in the fifth. Sierra hit a one-out double and was waved home on Ryan Goins' single, but Gardner charged the ball and threw it to catcher Chris Stewart, who tagged out the sliding Sierra.
Reynolds hit a leadoff homer off Rogers in the sixth, his 17th.
NOTES: Toronto grounded into inning-ending double plays in the sixth, seventh, eighth and ninth. J.P. Arencibia was the batter in the seventh and ninth. ... Blue Jays GM Alex Anthopoulos said Gibbons will return in 2014, even though the team is in last place. ... Gibbons said RHP Josh Johnson (strained right forearm) won't return this season. ... Toronto RHP Chien-Ming Wang cleared waivers and was sent outright to Triple-A Buffalo.