Moore (14-3) yielded just a second-inning single to Mike Napoli and hit to David Ortiz in the seventh but otherwise baffled the Bosox, allowing one walk and striking out four en route to his first career shutout.
The left-hander fired 74-of-109 pitches for strikes and lowered his ERA to 3.17 as the second-place Rays reeled off their sixth straight win and 18th in 20 contests, pulling within one-half game of American League East-leading Boston.
James Loney drove in a pair of runs with an RBI single and a sacrifice fly in the triumph.
Rookie Brandon Workman (0-1) suffered his first career loss in his third start, giving up two runs on seven hits over seven innings as the Red Sox dropped the opener of a key four-game series.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys