Red Sox starter John Lackey's throwing error on a pickoff attempt allowed Rodriguez and Escobar to move into scoring position. Evan Longoria then hit a sacrifice fly to center field to score Rodriguez and give the Rays a 2-0 lead.
Making his fourth start since having right-elbow reconstruction surgery in November 2011, Lackey gave up two runs on four hits and a walk while striking out two in 3 2/3 innings. He did not pitch in 2012.
Making his third outing of the spring, Matt Moore started for the Rays and pitched three scoreless innings before finding trouble in the fourth.
The Rays' southpaw walked Daniel Nava to start the inning. One out later, Ryan Sweeney singled to right before David Ross homered to left-center field to give the Red Sox a 3-2 lead.
"I felt like I was able to stay on top of my fastball much better," Moore said. "Pitched down in the zone. ... Felt like I made a good adjustment with my offspeed pitches in the middle of the at-bat, which felt nice."
Boston tacked on three more runs in the seventh, highlighted by Pedro Ciriaco's RBI single and Jeremy Hazelbaker's run-scoring double.
Ross' home run was his first hit of the spring.
Moore, who won 11 games in his rookie season in 2012, allowed three runs on four hits, walking two and striking out two.
"Minus the three-run home run, there were some hard-hit baseballs, but I was getting through the inning in 10-12 pitches. ... I was able to make some pitches I had to make," Moore said.
Josh Lueke continued his impressive spring for the Rays by pitching a scoreless 1 2/3 innings to move his ERA to 1.29 through four outings.
Up next: Lefty ace Jon Lester will take the long bus ride to Jupiter and start Monday's 1:05 p.m. ET contest against the Marlins. Will Middlebrooks, Mike Napoli, Jarrod Saltalamacchia and Jonny Gomes are expected to be in the starting lineup.
Bill Chastain is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.