Gaudreau set up goals by Sean Monahan and Michael Stone to give the Flames an early lead, then scored the second of two Calgary goals 50 seconds apart early in the second period.
"For us here in our own building, we wanted to get off to an early lead and play with the lead throughout the game," said Gaudreau, who leads the Flames in scoring with 52 points in 62 games.
"We came out flying, had a couple big goals there, and played with the lead the rest of the game. It was a huge win for us."
Mark Giordano and Kris Versteeg also scored for the Flames, and Brian Elliott made 19 saves for his 11th straight win to tie a club record set by Mike Vernon in the 1988-89 season.
"I think any time you're mentioned in the same sentence as a guy like Mike Vernon, it's an accomplishment," said Elliott, who has gone 15-1-1 dating to Jan. 26. "It's pretty cool to be able to have done that, but for right now, it's the two points we got and they didn't."
The Flames moved into third in the Pacific Division past the Edmonton Oilers, who will host Los Angeles on Monday.
Anze Kopitar and Nic Dowd scored for the Kings, who remain six points behind the Nashville Predators for the final Western Conference wild card.
"You break it down, they played well and we weren't good enough tonight," Kings forward Jarome Iginla said. "Those quick strikes hurt us."
Goalie Jonathan Quick started for the Kings but was replaced by Ben Bishop at 11:36 of the first period after giving up two goals on seven shots. Bishop stopped 14 of 16 shots.
Monahan opened the scoring at 8:15 of the first period when he took a drop pass and fired a shot to the top corner over Quick's glove hand.
Only a few minutes later, Quick made a nice play to stop a shot from the slot by Monahan. The Flames kept the puck in L.A.'s zone, and Stone took a pass from Gaudreau before blasting a shot past Quick at 11:36 to put the Flames up 2-0.
Kopitar scored for the Kings with 1:20 to play in the opening period when he redirected a pass from Dustin Brown past Elliott.
Giordano put the Flames back up by a pair 47 seconds into the second period before Gaudreau converted on breakaway chance 50 seconds later by lifting a backhand shot over Bishop's left pad.
Dowd cut Calgary's lead to 4-2 at 5:31 by redirecting Paul LaDue's point shot past Elliott for his first goal since Dec. 22, ending a 33-game drought.
The Kings couldn't get much going after that as the Flames tightened up defensively to record their second straight win and 12th in their past 13 games.
Versteeg rounded out the scoring with an empty-netter with 24.8 seconds to play in the game.
Prior to the game, the Flames announced Micheal Ferland would not play against the Kings and that he was under observation for the mumps. ... Alex Chiasson took Ferland's place on a line with Monahan and Gaudreau and had an assist in the first period for his fifth point in his past seven games. ... Ferland's absence made room for Curtis Lazar to make his debut for the Flames. Lazar had yet to get into the lineup since being acquired from the Ottawa Senators on March 1.
Kings: Travel to Edmonton for a game Monday night.
Flames: Start a three-game trip at Washington on Tuesday night.