With Bobby Abreu delivering the game-breaking hit, a two-run double in the sixth inning, Weaver (4-2) struck out eight while yielding only four hits and two walks across eight innings. Brian Fuentes worked a scoreless ninth for his 13th save.
"He's quietly having a terrific season," Angels manager Mike Scioscia said. "That was his fourth win, but he's pitched much better than that."
Weaver's 2.36 ERA is surpassed among American League pitchers only by Zack Greinke's outlandish 0.84 for Kansas City. Weaver has yielded one or fewer runs in four of his past five starts and five of his past seven.
Weaver, who threw 103 pitches, called it "a little frustrating" that he was unable to go for his third complete game of the season, but he said he understood Scioscia's reasoning.
The manager alluded to "a little knot" in Weaver's back that he was overcoming, adding that he didn't want to risk stretching him out to "115 to 118 pitches to end it -- or to have to bring Brian in with a couple men on base."
"With the guys we have out there [in the bullpen], the way they've been throwing, I'm more than happy to give it up," Weaver said. "You want to keep battling, but you have a lot of trust in those guys."
With Gavin Floyd tossing up zeros for the White Sox, Weaver fell behind in the fourth inning when he surrendered two of Chicago's four hits. But the Angels manufactured a three-run rally in the sixth inning against Floyd, who fell to 3-5 with the loss.
Abreu's two-run double to right gave Weaver the lead, and Juan Rivera -- raising his average to .293 with two hits -- also doubled home a run.
Ending an 0-for-15 drought, Jeff Mathis opened the inning against Floyd with a single to the hole at shortstop. The athletic catcher advanced as Chone Figgins extended his hitting streak to a career-high 15 games with a single to right.
After Abreu, who ended his homerless drought on Tuesday night, drove in Mathis and Figgins, Rivera's drive to the left-center gap cashed in Abreu.
"The double helped Weaver," Abreu said of the curveball he smacked into the right-field corner. "He was throwing a heckuva game."
Scioscia felt his offense had a productive night but was frustrated by several shots handled by the White Sox. Crisp singles by Abreu and Vladimir Guerrero had the Angels threatening in the fourth, but first baseman Paul Konerko backhanded Kendry Morales' hard grounder to take a hit and two RBIs away.
Maicer Izturis lined into a double play in the fifth with Rivera, after a single, running.
"I think we had a good night in the batter's box," Scioscia said. "Fortunately, Weaver was up to the challenge on the mound."
Weaver now owns a 0.51 career ERA against Chicago and is 4-0.
The Sox finally broke through against the crossfiring Weaver in the fourth when Jermaine Dye singled, took third on Jim Thome's double and scored on Konerko's sacrifice fly.
"Weaver always seems to be on his game against us," catcher A.J. Pierzynski said. "We had the one chance when we scored the one run and could have maybe got more, but other than that, he was pretty good. He didn't make too many mistakes and threw the ball well."
White Sox manager Ozzie Guillen was impressed.
"This kid commanded the ball real well," Guillen said, "and we couldn't do too much about it."
Weaver came out with guns blazing, striking out five of the first nine men he faced. He'd allowed only one hit before Dye's single in the fourth.
"He had everything working," Mathis said following the breezy, two-hour, five-minute affair. "He was putting his fastball in good spots, and his slider was good. We brought the change into it in the middle innings. Jered was really good tonight."
Meanwhile, in a California League game in Lancaster, another Angels starter -- Kelvim Escobar -- was showing he's ready to rejoin the rotation.
Pitching for high Class A Rancho Cucamonga, Escobar was dominant across six shutout innings, yielding one hit and no walks while striking out eight hitters. He retired the last 12 men and threw 76 pitches.
Escobar, who underwent right shoulder surgery on July 29, is eligible to come off the 60-day disabled list on June 4.
Weaver, a high school basketball star in Simi Valley, Calif., and a die-hard Lakers fan, was beaming in the afterglow of triumphs by his two favorite Southern California teams.
"One more to get," he said, referring to the Lakers' NBA conference finals matchup with the Nuggets.
The road is much longer for the Angels, who are getting increasingly comfortable with Weaver at the wheel.