Bonino added an assist for four points and Palmieri's tally late the second period broke a 4-4 tie, putting the Ducks ahead to stay.
Jonas Hiller surrendered four goals on 34 shots but still emerged with the victory.
The Kings' Jonathan Quick, however, was yanked after only 5 1/2 minutes in net, during which he coughed up a pair of goals to Bonino on only three shots.
Jonathan Bernier relieved and turned aside 17-of-21 shots the rest of the way.
Dustin Brown scored twice for Los Angeles, which dropped a second straight.