Moore (12-3), who could still become an All-Star as an injury replacement, gave up five hits, two walks and struck out six over 6 1/3 innings.
"He certainly pitched like an All-Star," Tampa Bay manager Joe Maddon said. "There's still an opportunity, possibly. We'll see how it all shakes out the next couple days."
The Rays have won seven of eight, improving to a season-best eight games (48-40) over .500. Fernando Rodney, the third Tampa Bay reliever, pitched the ninth for his 19th save -- completing the six-hitter.
"We're playing great right now," Moore said. "On my night, it's just keep the ball rolling."
Sale (5-8) allowed three runs, six hits, one walk and had nine strikeouts in seven innings for the White Sox, who have lost eight of 10. The left-hander, despite giving up just 20 runs -- 17 earned -- over 49 1/3 innings, is 0-6 in his past seven starts.
"I think it would be frustrating for anybody," Sale said. "I think everybody here is frustrated, not just me. You've just got to keep trugging along. When you start getting down on yourself, that's when things get even worse than they are."
Sale and Moore hooked up in a pitchers' duel on May 28th of last year at Tropicana Field, a 2-1 White Sox victory in which the pair combined for 25 strikeouts (Sale 15, Moore 10).
Tampa Bay took a 2-0 lead in the second when Luke Scott got the first extra-base hit this season by a left-handed hitter off Sale, an RBI double, and Desmond Jennings had a run-scoring grounder.
"You can't expect him to blank everybody," White Sox manager Robin Ventura said. "He's pitching fine, competing and doing all the things you want him to do. It's just when you don't get any runs, it seems like it's wasted."
Wil Myers opened the sixth with an infield single, a liner that went off Sale's glove, and advanced two bases to third on a nifty baserunning play when Jose Lobaton hit a one-out grounder that third baseman Conor Gillaspie had to move toward second to field. The Rays rookie then scored to make it 3-0 on a single by Sam Fuld.
On "Turn Back the Clock Night," the Rays, an expansion team in 1998, wore hypothetical uniforms from 1979, while the White Sox used 1972 replica home uniforms.
"It definitely reeks of the mid-1970's, and I like that," Maddon said. "I think they're very tasty."
Injured White Sox reliever Jesse Crain was also picked to the AL All-Star team with Sale before the game. Tampa Bay's lone selection was infielder-outfielder Ben Zobrist.
Moore has not allowed a run in his past 17 1/3 innings. ... Tampa Bay LHP David Price (2-4) will make his second start Sunday since returning from a strained left triceps. The 2012 AL Cy Young Award winner, who allowed three hits over seven innings in an 8-0 win over Houston on Tuesday, will face Chicago LHP John Danks (2-5). ... White Sox RHP Jake Peavy, on the DL since June 6 with a broken left rib, is scheduled to pitch in a simulated game on Tuesday. ... Maddon decided to keep left-handed hitting 1B James Loney, who has a career-high 16-game hitting streak, out of the lineup against Sale. Loney entered as a defensive replacement in the eighth, but didn't bat.. ... Chicago OF Dewayne Wise (strained right oblique) is set to start a minor league rehab assignment with Triple-A Charlotte.