It was his first shutout of the season and his ninth in the NHL.
Forsberg stopped Ridly Greig's backhander on a breakaway to keep it 1-0 at 11:21 of the third period.
The 32-year-old, who played 142 games for the Senators from 2021 to 2025, signed a two-year, $4.5 million contract with the Kings in free agency on July 1.
Alex Laferriere scored for the Kings (10-5-4), who have won four straight and have won the first four games of a six-game road trip.
Linus Ullmark made 17 saves for the Senators (9-6-4), who had their seven-game point streak snapped (4-0-3).
Laferriere tipped a Mikey Anderson point shot from in front to give the Kings a 1-0 lead at 9:19 of the first period.
Kings defenseman Drew Doughty left the game late in the second period after blocking a shot off his foot.
