Shoemaker (3-2) allowed four runs in the first two frames, including a two-run homer by Jose Abreu in the first, but nothing after that as he went 6 1/3 innings and struck out a season-high nine.
"Really, no big adjustments," Shoemaker said. "Just staying aggressive, attacking the zone. You've just got to go out there and keep pitching, keep trying to get outs and put up zeros, and then your team picks you up, which is great."
Said Angels manager Mike Scioscia: "He had to earn every out. He made some pitches when he had to, got some big strikeouts when he had to. That lineup over there, they've got some guys that are swinging the bat as well as anyone in the league. He got into the seventh inning, which is a good sign."
The night didn't go as smoothly on the mound for the White Sox. Starter Miguel Gonzalez saw his four-run lead evaporate in the second, with Albert Pujols tying the game, 4-4, with a two-run single. Martin Maldonado's RBI single in the sixth closed the book on Gonzalez and gave the Angels a 5-4 lead they wouldn't relinquish.
"Guys are not swinging at my pitches," Gonzalez said. "They're being a little more patient, and I'm falling behind. If we can fix that, then we'll be OK. Things will change."
Chicago starters are now 0-8 with a 6.36 ERA over the last 11 games, with opponents hitting .294 over that span. But it didn't get any easier for the bullpen.
Anthony Swarzak relieved Gonzalez and took a streak of 19 2/3 scoreless innings to the mound, but consecutive singles by Maldonado and Cameron Maybin preceded Trout's three-run blast, ending the longest scoreless streak in the Majors and wrapping up another four-spot for the Halos.
"I don't care what kind of year you're having, what streak is going on or whatever," Swarzak said. "If you're giving up three consecutive hits to three consecutive batters, that's an issue."
Maldonado picked up his third RBI of the game with a single in the seventh in yet another four-run outburst for the Angels, who held off a late charge by the White Sox to climb above .500 for the first time since May 3.
"The character of this club is unbelievable," Shoemaker said. "Great clubhouse, great team. If we can just put it together and play well and everybody does their part, it can be really good."
MOMENTS THAT MATTERED The Machine continues to rage: A night after hitting the game-winning RBI, Pujols once again came through for the Angels, drilling a two-run single to tie the game, 4-4, in the bottom of the second. He has 32 RBIs on the season.
I'll take a Martin. Actually, make it three: Maldonado came through at the plate for the Angels, notching a season-high three RBIs. His second RBI -- a dribbler down the third-base line -- plated Ben Revere with the go-ahead run.
SWARZAK'S RUN ENDS
Swarzak's run of excellence, bordering on mind-boggling, came to a temporary end in the sixth inning when the right-hander allowed three runs on three hits. Entering the night, Swarzak had allowed two hits and one walk in his last 50 batters faced and had a scoreless streak of 19 2/3 innings come to an end. He was the only pitcher remaining the Majors to work 19-plus innings without allowing a run in 2017.
"He got a little tight on the ball he hit to the wall in right-center [in the fifth inning]. Running, he felt OK. In the [sixth], he was a little tight so we took him out as a precaution. Right now, it's day to day." -- Scioscia, on pinch-running for Pujols after he walked in the sixth inning
"We've lost 14 of the last 15 here? I wasn't aware of that. OK. I don't know. I'll have to think about that a little." -- White Sox manager Rick Renteria, on Chicago's struggles in Anaheim
UPON FURTHER REVIEW
Abreu drilled a ground ball into a 6-4-3 double play during the fifth inning. He was initially ruled safe at first, but was called out after review. The Angels have won seven of 10 challenges this season.
WHAT'S NEXT White Sox:Dylan Covey (0-3, 7.98 ERA) makes his seventh start of the season, fifth on the road and his first career start against the Mariners when Chicago opens a series in Seattle on Thursday at 9:10 p.m. CT. The rookie felt as if he had a breakthrough in his last trip to the mound against the Padres, striking out a career-high nine over 4 1/3 innings..
Angels: The Angels will open a 10-game road trip, beginning with the opener of a three-game series against the New York Mets at 4:10 p.m. PT on Friday. Ricky Nolasco (2-2, 4.34 ERA) will make his ninth start of the season. He's 6-8 with a 5.03 ERA (139 2/3 innings and 78 earned runs) in 26 appearances (24 starts) against the Mets.