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.