The backup to Nikolai Khabibulin, Weekes stopped 12 shots in the first period, 10 in the second and nine in the third for his seventh career shutout.
Prospal got the game's lone goal 38 seconds into the final period when he came down the right side on the rush and put a wrist shot from about 10 feet over goalie Felix Potvin's left shoulder.
Fredrik Modin and Martin St. Louis picked up assists for the Lightning, who improved to 11-5-1 all-time against the Kings. They are 8-1-1 in Los Angeles, winning the last four meetings here since a 5-2 setback on Nov. 6, 1997.
It was the first win of the season for Tampa Bay, which had dropped its first three games and seven in a row dating back to the 2000-01 campaign.