The 37-year-old, who joined the west coast club after leaving AC Milan at the end of last season, tapped in from close range to give his side the lead in the 53rd minute.
Kansas City’s Erik Thommy equalised to take the game to extra time, when Mexican defender Omar Campos and Sierra Leonean forward Kei Kamara scored to seal the win.
For Giroud, who retired from France duty in July as the nation’s record scorer with 57 goals, it was important after August’s Leagues Cup final defeat by Columbus Crew.
“I came here to win trophies and to play that kind of final,” said Giroud.
“I was really sad and disappointed for the team regarding the Leagues Cup.
“We’ve been in a difficult run these last weeks, but we stuck together, and we really wanted to win this trophy for the fans, for the club.
It does matter for me, for the boys, and I’m so happy to score in this final and to help the team.”
The victory keeps LAFC’s hopes of completing a domestic league and cup double alive.
Only three teams have done the double, with LA Galaxy doing so in 2005 after Chicago Fire and DC United achieved the feat.
LAFC sit fourth in the MLS Western Conference standings and are expected to seal a play-off spot.
