"We needed a victory like this," Trout said. "We never stopped fighting, had great at-bats, put some things together and we came out and got the W."
Chicago starter Chris Sale took a three-hit shutout into the eighth and the White Sox held a commanding 5-0 lead.
Erick Aybar led off the eighth with a double and scored on Chris Iannetta's single, and after an error and another single loaded the bases, Trout hit a full-count changeup over the left-center field fence to tie it.
Aybar stroked a two-out RBI single later in the inning, and Ernesto Frieri closed it out for the Angels.
The grand slam was Trout's only hit of the game. He has a hit in 14 of his past 15 games and has reached base in 26 of his past 27.
"I was just [using a] two-strike approach," he said. "I wasn't trying to hit a home run, but it went out. The first two swings I was thinking home run, that's why I fouled them off. You try to get too big in situations you lose mechanics of your swing. That 3-2 pitch I just told myself to stay up the middle and put the ball in play."
Sale was all but unhittable for the first seven innings. The 25-year-old left-hander didn't allow an Angels runner past second base to that point and appeared firmly in control.
The Angels finally figured him out in the eighth. After Aybar doubled and scored on Iannetta's single, Collin Cowgill reached on an error by shortstop Alexei Ramirez.
Howie Kendrick poked a single through the right side to bring Trout up with the bases loaded and no outs, and the Angels outfielder belted the tying homer to whip the home crowd into a frenzy.
"I don't really remember much from it," Sale said. "I just remember that ball going over the fence and wanting to rip my own head off.
"That's why he's the best in the league. I can't really say too much about a guy like that hitting a home run off of you."
Trout's blast, his 12th homer of the season, ended Sale's night. The Angels weren't done.
Hamilton and C.J. Cron hit back-to-back two-out singles off reliever Jake Petricka (0-1), and Aybar poked a 2-2 fastball into center field to score Hamilton and give the Angels their first lead of the game.
"We never quit," Trout said. "Down 5-0 we could easily just put our heads down and stop fighting, but you never know. Everybody contributed. It was a team win for sure."
Cory Rasmus (1-0), who was recalled from Triple-A Salt Lake before the game, pitched two scoreless innings of relief for the win.
The White Sox opened the scoring in the third on back-to-back RBI singles by Adam Eaton and Gordon Beckham, and Alejandro De Aza smacked an RBI double into the right-center field gap in the fourth to extend their lead to 3-0.
Adam Dunn led off the sixth with a solo homer deep to center field, his third consecutive game with a home run.
The blast, which easily cleared the 396-foot marker in center, ended the night for Angels starter Matt Shoemaker.
Shoemaker surrendered nine hits and four runs, three earned, while striking out six.
De Aza brought home another run on a sacrifice fly later in the sixth to give the White Sox a seemingly comfortable 5-0 lead.
"When you put as much effort, and kind of your heart and soul into a game and it unravels like that, more than anything I'm just kind of disappointed," Sale said. "I threw a good pitch to a better hitter, and things happen."
Chicago rookie Jose Abreu went 0-for-4 with four strikeouts, the first four-strikeout game of his career. ... Hamilton's fifth-inning single was the first hit by a left-handed batter against Sale this season. ... The Angels optioned outfielder Grant Green to Triple-A Salt Lake before the game. ... The Angels celebrated Albert Pujols' joining the 500-homer club in a pregame ceremony. Tony La Russa was among the speakers.
Copyright by STATS LLC and The Associated Press
ANAHEIM, CA - JUNE 07: Matt Shoemaker #52 of the Los Angeles Angels throws a pitch in the first inning against...