Glasnow (8-5) allowed one run and three hits over six innings in a season-high, 103-pitch outing. Boston was 0 for 7 with runners in scoring position against the 6-foot-8 right-hander.
“It felt good, it felt timed up, and that was a good feeling for sure,” Glasnow said, “I hope I can do that a lot more.”
Glasnow has gone six or more innings in nine straight starts.
“You got to have some dominant starting pitching to get where you want to get,” Rays manager Kevin Cash said. “Certainly we want to play well here in September, and he's a big part of that. Today was a pretty dominating performance.”
After Robert Stephenson and Colin Poche went an inning each, Pete Fairbanks worked the ninth to get his 20th save and complete a five-hitter.
Brandon Lowe and Isaac Paredes hit solo home runs off Boston starter Nick Pivetta (9-8).
Tampa Bay took two of three in the series, and have won 14 of 15 at home against the Red Sox.
Boston struck out 17 times in the game, and finished with 47 strikeouts during the series.
The Rays hold a commanding lead for the top AL wild card. Boston started play five back of the final wild-card spot.
"We're going home now,” Boston manager Alex Cora said, “If we want to gain ground, we've got a chance now. We've got the best team in the American League (Baltimore) coming up. We've got the Yankees and then Toronto and Texas. This week is going to be huge for us.”
Boston rookie Triston Casas singled in the ninth to extend his career-best hitting streak to 13 games
Tampa Bay took a 2-1 lead on Paredes' shot in the fourth. It gave him three homers in 10 at-bats when facing Pivetta.
Lowe tied it at 1 with a third-inning homer. It was his third long ball in 13 at-bats against the right-hander.
Harold Ramírez, on his 29th birthday, ended Pivetta's night with a run-scoring double that made it 3-1 in the fifth.
Connor Wong had an RBI triple in the third that put Boston up 1-0. Rays shortstop Osleivis Basabe help keep Wong from scoring when he made a diving grab on Rafael Devers' grounder up the middle with the infield in.
Pivetta gave up three runs and seven hits over 4 2/3 innings in his first start since Aug. 15. He was coming off a three-inning save on Saturday,
“It's been great,” Cora said. “I told him on the mound, ‘I said ’thank you bro.' What he did today, that was good to see,”
Devers had a first-inning double off a catwalk. As right fielder Luke Raley drifted back to the warning track, the high fly struck the overhanging structure and landed in shallow right.
Red Sox: OF Alex Verdugo (left hamstring tightness) missed his third straight game, but manager Alex Cora is hopeful he will be back Friday. … LHP John Paxton is getting extra rest and is scheduled to start Tuesday against the New York Yankees on 10 days of rest. … RHP Corey Kluber (right shoulder inflammation) will pitch for Double-A Portland on Friday.
Rays: SS Taylor Walls (groin) sat out the second consecutive game, but is expected to start Thursday.
Red Sox: RHP Tanner Houck (4-8) is set to start Friday night in the opener of a three-game home series against AL East-leading Baltimore.
Rays: RHP Zack Littell (3-4) was scheduled to face Seattle RHP Luis Castillo (11-7) at home Thursday night.
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