"I had a pretty good idea," Lester said when asked if he knew he left after 18 up and 18 down. "I pitched the whole time in the wind-up. ... You got the scoreboard right in your face. It's too hard to miss."
Cobb, meanwhile, was unable to contain a lineup stocked with Red Sox regulars. The right-hander, who entered the day without allowing a run in his previous three starts, surrendered four runs (three earned) on six hits over 5 1/3 frames. He also struck out five and walked one.
Manager Joe Maddon wasn't concerned his team might be the victim of a perfect game, but he thought it might happen.
"It was there for them," Maddon said. "We have a lot of practice rebounding from those kinds of events. From that perspective, it was not that big of a concern. We've been able to go to the playoffs, win 90-plus games in spite of those moments. If anybody can deal with a perfect game or no-hitter, we can.
"I'm certain that we're going to flirt with that at another time during the course of the season, but if you don't win it, it's just another loss and you move on from there. We've taught ourselves some lessons."
Lester is in line to start Opening Day for the Sox on April 1 in New York, but the lefty said after the game he hasn't been told anything. Manager John Farrell said before the game he had no announcement.
Jonny Gomes went 3-for-3 and drove four of the Sox's runs.
Anthony Carter and Joel Hanrahan kept the perfecto going in the seventh and eighth.
The Sox carried the perfect game all the way until one out in the ninth, when Jason Bourgeois hit an infield single off Marco Duarte. Cole Figueroa then got Tampa Bay on the scoreboard with a ground-rule double to right field.
Bourgeois' single was the first time he broke up a perfect game.
"I've never done that before," said Bourgeois. "It's not my game to hit the ball out of the park, and it's always the little things that make it happen. I'm glad I could shed some light on the game for us.
"Around the seventh inning, we started to grip the bats a little tighter. I'm glad we got something, and got something on the board.
In 20 Grapefruit League innings spanning five starts, Lester has surrendered just two runs (0.90 ERA) and six hits. With six strikeouts Sunday, he now has 16 this spring.
Up next: The Rays have an off-day Monday before travelling to Lakeland on Tuesday to take on the Tigers at 1:05 p.m. ET. Look for Ben Zobrist to get his first Grapefruit League action following his World Baseball Classic experience. Jeremy Hellickson will make his fourth appearance, and his second start, for the Rays.