Billy Butler and Alex Gordon had RBI singles in the first inning, and Joey Gathright scored on an error in the second to give Kansas City a quick 3-0 lead.
The Royals extended their win streak against the Angels to six in a row, five of those last year.
Although Weaver (2-2) wasn't sharp, he managed to last six innings. He allowed 10 hits - including five singles in the first inning - and the three runs. One of those was unearned, with a fielding error by the Angels' right-hander letting Gathright score.
Relievers Ramon Ramirez, Jimmy Gobble and Leo Nunez shut out Los Angeles until Joakim Soria took over in the ninth to earn his fifth save in five chances.
Soria pitched a 1-2-3 ninth with two strikeouts, and has retired the last 17 batters he's faced, eight on strikeouts.
Los Angeles slugger Vladimir Guerrero left after eight innings with a bruised right index finger, hurt on a swing in the seventh. The injury was not considered serious and Guerrero is day-to-day.
Mark Teahen went 3-for-4 for Kansas City, and Gathright 3-for-5.
The Angels narrowed the gap to 3-2 with a two-out rally against Meche in the fourth inning. Garret Anderson singled, Torii Hunter doubled, then both scored on Casey Kotchman's line drive single to center.
Gathright scored from second base when Weaver, covering first base on Mark Grudzielanek's grounder, dropped the toss that would have been the third out.
Meche got no decision in his first start of the season, giving up three runs in six innings of a 5-4, 11-inning victory over Detroit. He allowed six runs on 5 2-3 innings his next time out, a 6-4 loss to Livan Hernandez and Minnesota, then was tagged for five runs in a rematch against Hernandez and the Twins in a 5-0 defeat.
Weaver was coming off his rockiest outing of the season, giving up five runs in 5 2-3 innings of an 8-5 loss to Seattle.
Notes: Los Angeles optioned OF Reggie Willits to Triple-A Salt Lake and selected the contract of RHP Alex Serrano from Salt Lake. ... The Royals came into Wednesday's game with their starters sporting the AL's best ERA (3.10) and their defense topping the league in fielding percentage (.994). The Angels were leading the majors in batting (.301) and runs (20). ... Los Angeles reliever Chris Bootcheck (strained oblique) will continue his rehab in the minors and may be back for the Angels' next road trip, which begins Tuesday in Boston. ... Meche started the game with a 6.07 ERA in 10 career starts against the Angels in Anaheim, but the win gave him a 7-1 record in their stadium.