Joyce completed the rally by lining a hit to right field for a 4-0 advantage, and the American League wild card-leading Rays never looked back.
Matt Moore (10-7) held an opponent to two runs or less for the seventh consecutive start, yielding a pair of runs on five hits over 6 1/3 innings. He allowed two walks and struck out five.
Greinke was roughed up for six runs on seven hits in six frames for the Angels, who fell to 5-13 in August.
Mike Trout and Albert Pujols socked solo homers for Los Angeles.