Glasnow (3-1) gave up five hits and two walks, dealing the Athletics their third loss in four games after a 13-game winning streak.
The big right-hander threw 98 pitches while dealing with a cramp in his right leg that he said made him feel ''weirdly uncomfortable'' in the dugout. Glasnow said he had to ''grind through it.''
Given his dominant results, Glasnow had second thoughts about mentioning the cramp.
''Can we just forget I said that?'' he said. ''I feel like an idiot that I even said that.''
Diego Castillo got the final two outs to earn his sixth save in eight chances.
Mike Zunino's fifth home run, a towering blast off the Tropicana Field catwalk, broke a scoreless tie in the fifth inning.
''Sometimes when you take an easy swing and try not to do too much, that's what happens,'' said Zunino, who is on a six-game hitting streak.
The Rays made it 2-0 after Oakland starter Cole Irvin's only walk and back-to-back singles by Randy Arozarena and Manuel Margot.
Oakland's only serious threat against Glasnow came in the sixth when Mitch Moreland bounced a two-out double through an infield vacated by a shift. Matt Chapman followed with a sharp single, but Glasnow got Seth Brown to ground out.
In 13 starts going back to Aug. 19, Glasnow is 8-1 with a 2.37 ERA and 120 strikeouts in 79 2/3 innings.
Irvin (2-3) struck out eight in six innings while giving up two runs on seven hits.
''If we had scored five, we'd be talking about how good he pitched,'' Oakland manager Bob Melvin said. ''Obviously we just couldn't solve Glasnow.''
After averaging 6.2 runs during their winning streak, the A's have totaled six runs in their last four games.
Athletics: OF Chad Pinder (sprained left knee) is running again and feeling well, but is not close to a return, Melvin said. Pinder last played on April 4.
Rays: RHP Pete Fairbanks (strained rotator cuff) threw at Tropicana Field on Wednesday and will face hitters this weekend in hopes of being activated for next week's trip.
Arozarena took the high road after Kevin Kiermaier called him off on a flyball Tuesday night, a curious decision that left Arozarena shaking his head. Kiermaier, a three-time Gold Glove winner, ran over from center field, called off Arozarena to make the catch, then threw home in an unsuccessful attempt to get a tagging Matt Olson.
''I was ready to make the play, but he's the center fielder and he took over. It could happen to anybody,'' Arozarena said. ''Some things you can't control in a game. There's no problem between us. First and foremost, I'm glad we won the game.''
RHP Chris Bassitt (2-2, 4.13 ERA) will pitch the series finale for Oakland on Thursday against LHP Shane McClanahan, who will become the first pitcher in major league history to have appeared in a World Series game before making his regular-season debut. He pitched the ninth inning of Game 3 against the Dodgers on Oct. 3, 2020.
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