Smoak went 3 for 3 and scored twice.
Francisco Liriano (2-2) allowed one run and four hits in five-plus innings. He walked four and struck out six.
Liriano was replaced by Joe Biagini after a walk and an error put Rays at first and second with none out in the sixth. Tim Beckham struck out trying to bunt, pinch hitter Brad Miller went down swinging and pinch hitter Logan Morrison grounded out.
Matt Andriese (1-1) gave up four runs in seven innings.
Rays outfielder Steven Souza left the game in the seventh after being hit on the back of the left hand by a pitch from Biagini. X-rays were negative and Souza is day-to-day.
Toronto took a 1-0 lead in the first when Jose Bautista scored from second on a throwing error by Beckham from shortstop. The inning ended one batter later when Kendrys Morales tried to score from second on Smoak's single, but was thrown out by center fielder Kevin Kiermaier.
Beckham slid around the tag of Blue Jays catcher Russell Martin to score the tying run in the second, coming home on Jesus Sucre's single to right. Plate umpire Sam Holbrook initially ruled Beckham out, but the call was overturned following a replay review.
The Blue Jays reclaimed their lead in the fourth on Martin's RBI double, then widened their edge in the seventh when Smoak connected off Andriese. The homer was Smoak's fourth.
Rays: RHP Jake Odorizzi (left hamstring) is expected to come off the 10-day DL to start at Miami on Monday.
Blue Jays: LHP J.A. Happ (elbow) made between 30 and 40 throws on flat ground in the outfield and reported no pain. Happ last threw on April 20. ... 3B Josh Donaldson (right calf) and SS Troy Tulowitzki (right hamstring) both took batting practice and played catch before the game.
Rays: RHP Chris Archer (2-1, 3.94) has 20 career starts against Toronto, more than any other opponent. He's 6-4 with a 3.17 ERA in those outings.
Blue Jays: RHP Aaron Sanchez (blister) is expected come off the 10-day DL and start Sunday's series finale. Sanchez (0-0, 4.38) allowed five runs and seven hits in 5 1/3 innings in his previous start, April 14 against Baltimore.
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