Weaver, left open at the inside edge of the right circle, beat Chicago goalie Nikolai Khabibulin high on the glove side after taking a pass from Alexander Frolov.
Patrick O'Sullivan had a goal and assist for the Kings, 4-1-1 in their past six games following a five-game losing streak.
Patrick Sharp also scored for Chicago, which dropped its fourth straight.
Los Angeles defenceman Rob Blake left the game after the first period with an undisclosed injury.
Kings goalie Mathieu Garon made 26 saves. Khabibulin stopped 30 shots.
O'Sullivan, a rookie, extended his point streak to seven games. He has two goals and eight assists during the span.
Sharp scored for the third straight game and has seven goals in his past eight games.
Both the Kings and Blackhawks have been eliminated from playoff contention. Chicago will miss the post-season for the eighth time in the past nine seasons.
O'Sullivan opened the scoring during a five-on-three power play midway through the first period, beating Khabibulin on the glove side with a shot from the low edge of the right circle.
Khabibulin had stopped a Kings' attack seconds earlier. The puck landed in the right corner, where Mike Cammelleri passed it to O'Sullivan.
Sharp tied it at one with a power-play goal at 6:10 of the second. He slipped past the Kings defence on right wing, broke in on Garon and fired a shot over his shoulder on the stick side.
Los Angeles' Anze Kopitar hit the right post with a backhand shot with 5:40 left.
Notes: Chicago RW Peter Bondra sat out his fourth game with a shoulder injury. Blackhawks C Jeff Hamilton missed Friday's game with a sore back. ... Los Angeles C Derek Armstrong missed his seventh game with a knee injury. Kings D Lubomir Visnovsky sat out with an undisclosed injury and LW Noah Clarke was a healthy scratch.