Bedard allowed three runs and six hits in 4 1-3 innings. The left-hander is competing with Jake Odorizzi and Cesar Ramos to earn the fifth starter's job as a fill-in for the injured Jeremy Hellickson, who's recovering from elbow surgery and expected to miss the first six to eight weeks of the season.
The 35-year-old started 26 games and was 4-12 with a 4.59 ERA in 32 appearances with the Astros last season. Through four spring appearances, including two starts, he's 2-1 with a 7.15 ERA.
Rays manager Joe Maddon reiterated that the decision on the No. 5 starter will not be based on statistics alone.
''That's good because the ERA isn't good now,'' Bedard said.
''Every team's different. Sometimes they look at that and some teams don't,'' the former Orioles and Mariners pitcher added. ''They know it's spring and you're working on your stuff and just getting ready for the season.''
Mikie Mahtook hit a grand slam and Evan Longoria added a RBI single off Blue Jays starter Sean Nolin, who allowed five second-inning runs.
Four Tampa Bay relievers - Brandon Gomes, Grant Balfour, Joel Peralta and Brad Boxberger - combined to hold the Blue Jays hitless over the final 4 2-3 innings.
Kevin Pillar had a RBI triple off Bedard, then scored on the lefty's wild pitch for a 2-0 Toronto lead in the second.
''The pitching was spectacular,'' Rays manager Joe Maddon said, noting that Gomes, whose ERA remained at 0.00, was exceptionally sharp in 1 2-3 innings.
''He can't throw any better than he is right now,'' Maddon added.
Blue Jays: Nolin began with a perfect first, then walked three and allowed five hits during Tampa Bay's five-run second. His spring training ERA ballooned from 4.91 to 11.81 after the 1 2-3 inning outing.
Rays: Maddon said Bedard, Odorizzi and Ramos likely will all have at least one more appearance before a decision is made on the fifth starter.
''They're all three in the running,'' Maddon said, declining to say who has the inside track.
Bedard is trying to keep the competition in perspective.
''I'm not really worried. I'll just do what I can, and the decision's theirs,'' Bedard said. ''They'll do what they want, and that's fine with me.''
OF Wil Myers fouled a pitch off his right quad muscle during the Rays' split-squad game against the Pirates in Bradenton and was removed from the lineup for precautionary reasons.
Rays RH Juan Carlos Oviedo has resolved visa issues in the Dominican Republic and reported to camp Saturday. The former Royals and Marlins reliever spent the entire 2013 season on the disabled list recovering from Tommy John surgery. He will throw to some batters during Monday's off day, and a decision on when he'll pitch in a game will be made after that.
HOW'S IT GOIN'?
Toronto infielder Ryan Goins went 0 for 3, leaving him at .171 with no extra-base hits and no RBIs in 12 spring games. He spent time at second base and shortstop against the Rays, the positions he played last season while making his major league debut. The 26-year-old Goins hit .252 with two home runs and eight RBIs for Toronto after being called up in August.
Toronto manager John Gibbons said Goins might be pressing a bit.
''He knows he belongs in the big leagues,'' Gibbons said. ''He's a young kid.''
Most of Toronto's top hitters got the day off and didn't make the 2-hour drive. Jose Bautista, Jose Reyes and Melky Cabrera were among the regulars out of the lineup.
The Blue Jays haven't scored more than five runs for 12 straight games against big league squads.
''We haven't had that big outburst yet,'' Gibbons said.