Price allowed only three singles and walked two. His first seven outs all came by a strikeout, with two hits mixed in during that span.
Rays pitchers set a team mark by fanning 18 batters overall. Reliever Brandon Gomes struck out two in 1 1-3 innings and Cesar Ramos struck out two to end it.
Jennings homered on the first pitch of the game and got a career-high four hits. He drove in three runs, scored three, drew a walk and also stole a base from the leadoff spot.
The Blue Jays lost their fourth straight. They finished with five hits.
Price quickly matched his career high, striking out 12 through five innings. The left-hander didn't fan anyone in the sixth, but finished his outing with two more strikeouts in the seventh, matching and then passing Kazmir (2007) and Shields (2011) for the Rays' record.
Price improved to 9-1 with a 1.99 ERA in 11 career starts against Toronto.
Price didn't give his defense much to do in the first three innings. He did allow a pair of baserunners over that span: Mike McCoy led off the first with a walk and Jose Molina hit a two-out single in the second. Neither runner advanced.
McCoy was the first batter to make an out on a batted ball when he grounded to second in the third. McCoy, who walked again in the sixth, was the only Blue Jays batter not to strike out against Price.
Toronto's 18 strikeouts were a season-high, two more they had in a 3-1 road loss to the Los Angeles Angels on April 10. The franchise record is 19, set twice previously, both times in extra-inning games.
Jennings put Tampa Bay ahead right away, connecting against Brandon Morrow (9-9) for the first leadoff home run of his career.
After John Jaso's RBI double in the second scored B.J. Upton, Rodriguez hit a two-run shot into the second deck in left. Jennings made it back-to-back homers when he followed with another drive to left, his first multihomer game.
Morrow (9-9) allowed five runs and six hits in 5 1-3 innings. The right-hander has given up five homers in his past 10 innings. He lost for the fourth time in five starts.
NOTES: Blue Jays manager John Farrell (pneumonia) missed his third straight game, with bench coach Don Wakamatsu continuing to run the team. ... Rays RHP Kyle Farnsworth (elbow) was available after missing the previous four games. ... Toronto 1B Adam Lind, mired in a 3-for-21 slump, was held out of the starting lineup. ... Blue Jays RHP Jon Rauch (appendectomy) threw a bullpen session before the game, his second in three days. ... Toronto OF Rajai Davis (torn left hamstring) has begun jogging and taking batting practice at the team's minor league complex in Florida and could return before the end of the season. ... Tampa Bay RHP Wade Davis (8-7) faces Blue Jays LHP Ricky Romero (12-9) in Monday's series finale.