Steve Clevenger singled home Jemile Weeks, who had doubled to open the game, and Alex Gonzalez added a two-run single for the 3-0 Orioles cushion.
"It's the first game, I was trying to get my arm in shape and trying to throw strikes," Bedard said. "... My changeup was pretty good. I threw a lot of strikes with it and I was pretty confident with it. So that was the positive of the outing."
Heath Bell made his first appearance with the Rays after coming to the team in an offseason trade with the D-Backs. He pitched the second and surrendered a run on a walk to Weeks, an infield hit and a wild pitch.
Chris Tillman started for the Orioles and looked sharp in two innings of work, allowing no runs on one hit while striking out three. The right-hander is slated to be the Orioles' Opening Day starter.
"I felt good about my stuff," Tillman said. "Just wanted to work on fastball command for the most part.
"I think this early in spring, it doesn't really matter who you face. It's just going out there and getting the feel again; we face our own hitters the first two weeks and it isn't the same feel. So it was good to get the juices flowing again."
Left-hander Zach Britton followed Tillman and pitched a scoreless third before allowing a two-out RBI double to Jerry Sands that cut the O's lead to 4-1. Left fielder Justin Christian drove in another run for the Rays with an RBI single in the seventh. Yunel Escobar led the Rays with two hits, including a double.
Up Next: David Price will make his first appearance of the spring when he starts Saturday's 1:05 p.m. ET contest against the Pirates in Bradenton, Fla. Price has not yet been named the Opening Day starter, but this outing puts him in position to assume the duty. Also poised to see action are right-handers Mark Lowe, Brad Boxberger, Matt Andriese and Alex Colome, along with left-handers Jake McGee, Mike Montgomery, Enny Romero, and Jeff Beliveau.
Right-hander Jeanmar Gomez will start for the Pirates.
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.