Moore tossed seven innings and allowed just five hits and one run to stay undefeated and win his ninth straight decision, setting a Rays' franchise record.
The 23-year-old left-hander's start is the best by a pitcher that young since the 22-year-old Ruth went 8-0 in 1917.
"Historically speaking, it's been wonderful," Rays manager Joe Maddon told reporters. "I like the idea that he knows that he can get better. I love that. The accountability is tremendous."
Luke Scott homered in the fifth inning to break a 1-1 tie for Tampa Bay, and Matt Joyce added a blast in the sixth to lift the visiting Rays to a three-game series sweep.
In Cleveland, Justin Masterson enjoyed his own stellar pitching outing, striking out 11 batters in seven innings to lead the Indians to a 6-0 win over Seattle.
Masterson (7-2) allowed just three hits to overshadow Mariners ace Felix Hernandez who surrendered all six runs in a loss.
The American League Central-leading Indians (25-17) are red-hot, having won 17 of their last 21 games.
In the National League, Jeff Locke won a pitchers' duel against Lucas Harrell to help Pittsburgh pull out a 1-0 win against the Houston Astros.
Both starters completed seven innings, but Pirates infielder Pedro Alvarez smacked a home run in the fifth inning to provide the difference.
In Philadelphia, the Phillies rallied from a 2-0 deficit with late home runs to defeat Cincinnati 3-2.
After Chase Utley hit an RBI single in the eighth to make it 2-1, Erik Kratz and Freddy Galvis blasted back-to-back homers in the ninth to win it for Philadelphia.
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