Beavan (5-6) allowed only an RBI single to Matt Joyce in the first inning and then settled down to retire 18 of the next 20 batters.
The right-hander didn't issue a walk and struck out five for the M's, who took the series from Tampa Bay with two consecutive 2-1 victories.
Tom Wilhelmsen picked up his 10th save to secure the triumph.
Jesus Montero and Brendan Ryan provided RBI for the Mariners.
Moore (6-7) was tagged with the hard-luck loss, yielding a pair of runs on five hits over eight frames. The Rays hurler also didn't allow a walk and struck out seven in the loss.