Moore only allowed a solo home run to Jefry Marte and an RBI double to Andrelton Simmons over 6 2/3 innings to move to 5-5 on the season, earning his second straight win. Ryan Garton recorded the last two outs of the game to earn the save.
Morrison homered, singled in the go-ahead run and scored from first base on a double by Steven Souza Jr., that pushed the Rays' lead to 4-2.
Nick Tropeano started for the Angels and came away with a no-decision after allowing two runs on four hits in five innings. Fernando Salas took the loss, surrendering two runs in the sixth that proved to be the difference.
The Angels suffered their 50th loss in their 83rd game, the earliest they've reached that total since 1992.
"Our starting rotation has been really a project to get together," Angels manager Mike Scioscia said when asked why his team has yet to get going. "We have some guys back in our rotation now, but if you're not setting a tone with that starting pitcher, it makes it tough for the rest of your pieces to fall into place. Hopefully we'll get there."
Two hole works:Brad Miller hit a solo home run off Tropeano on a full count in the third inning, giving him 12 on the season. Clearly the Rays' shortstop likes hitting in the No. 2 spot in the lineup best, as 11 of his home runs have come while hitting second in the order. Miller's blast gave the Rays a 1-0 lead.
Having a rough go: Over his last 16 appearances, Salas has given up 18 runs on 22 hits and seven walks in 16 2/3 innings. Closer Huston Street (eight runs on 13 hits and five walks in his last 5 1/3 innings), setup man Joe Smith (a three-run inning in his second appearance since coming off the disabled list on Saturday) and middle reliever Mike Morin (4.93 ERA and now in Triple-A) have all struggled in the Angels' bullpen this season.
"The good thing is I'm healthy," Salas said in Spanish. "You just have to keep working. Things aren't really going my way right now."
Moore's 1-2 pitch: With two outs in the fourth, Moore was ahead of Jefry Marte 1-2 in the count. Unfortunately for the Rays left-hander, he hit Marte on the left knee with his next pitch. Moore followed that by walking Jett Bandy, then got burned when Simmons doubled home Marte to tie the game at 1.
Trop at The Trop: Tropeano pitched decently enough in his return to the rotation, giving up three earned runs or less for the ninth time in 11 tries in the Majors this season. Tropeano's only real mistakes were two changeups that he left up to Miller and Morrison, both of which resulted in home runs.
"Overall, I'm just kind of trying to get the rust out a little bit," said Tropeano, who recently missed about three weeks with a shoulder issue. "I had the adrenaline of just coming back here [with the Angels] for the first time, a lot of excitement. But too many balls early on. I ran my pitch count up."
QUOTABLE "Hopefully Ryan Garton has a smile from ear to ear and doesn't wipe it off all night." -- Cash, on Garton, who recorded his first Major League save
SOUND SMART WITH YOUR FRIENDS Moore has won consecutive starts for the first time since September 2013 vs. the Yankees and Blue Jays. He is 3-1 with a 2.43 ERA over his last five starts.
INJURY REPORT Marte, a corner infielder who has been used recently in left field, exited Monday's game early with a bruise on his left knee. Marte took a fastball there from Moore in the fourth, stayed in the game and homered in the sixth. He was replaced by Daniel Nava. Scioscia doesn't expect Marte to miss any additional time.
WHAT'S NEXT Angels:Tim Lincecum (1-2, 6.75 ERA) makes his fourth start of the season in the second of this four-game series, with first pitch set for 4:10 p.m. PT. Lincecum got roughed up in back-to-back starts from Angel Stadium, giving up a combined nine runs on 15 hits and six walks in 7 1/3 innings. His only start against the Rays came in 2013, when he threw seven innings of one-run ball.
Rays:Jake Odorizzi ( 3-3, 3.91) makes his 18th start of the season. The right-hander strives to pitch to contact. Last time out, that approach didn't work, as the Tigers fouled off 34 pitches, which resulted in an elevated pitch count and an early exit. He is 0-2 with a 5.14 ERA in three career starts against the Angels.