Chris Young and Seth Smith hit RBI doubles in the first and Scott Sizemore added a sacrifice fly, giving the A's a 3-0 lead.
Young, who also made a nifty catch in deep right field in bottom half of the inning, left the game before the start of the second as a precaution because of quad cramps.
A's left fielder Yoenis Cespedes, making his spring debut, went 0-for-2 with a walk. Center fielder Coco Crisp went 2-for-3 with a double and a run scored from the leadoff spot and first baseman Daric Barton went 2-for-2.
The Angels got three runs of their own in the bottom of the third against Brian Gordon, on an RBI double by Kole Calhoun, a run-scoring triple by Brendan Harris and a sac fly from third base prospect Kaleb Cowart.
In the top of the seventh, with most of the starters out of the game, A's infielder Darwin Perez broke a 4-4 tie with a two-run double off Andrew Taylor.
The only regular in the Angels' lineup was Mark Trumbo, though Vernon Wells could also win some playing time this year. Outfielder Matt Young, who's fighting for one of the Angels' final roster spots, went 2-for-3 and has five hits in his first six at-bats of the spring.
Travis Blackley, who has a good chance to stick with the A's as a long reliever, got through the first inning unscathed despite giving up three hits.
Enright, expected to start the season at Triple-A, threw 34 pitches before leaving with runners on the corners through two-thirds of an inning. He made no excuses about the weather. After an offseason spent tinkering with his delivery in hopes of finally getting some consistent time in the Majors, he saw this as a missed opportunity.
"It's hard to bite down that bullet today and have those results," Enright said. "The whole goal for me and [pitching coach Mike Butcher] was to push me over the top to be a regular big leaguer and not be a guy who's filling in here or being a sixth, seventh starter."
A's up next: Having completed two road affairs, the A's will play their first home game of the Cactus League season beginning at 12:05 p.m. PT on Monday, when the Indians visit Phoenix Municipal Stadium to face left-hander Andrew Werner, acquired from the Padres in the Tyson Ross trade. Fellow southpaw Hideki Okajima is also scheduled to pitch in the game, as are Jordan Norberto, Evan Scribner and 2011 first-round Draft pick Sonny Gray. Away from the mound, Jed Lowrie will take his turn at second base, and the A's will get their second in-game look at shortstop Hiroyuki Nakajima. Right-hander Justin Masterson will go for the Indians.