Winger Devante Smith-Pelly and center Ryan Getzlaf scored in the first period as the Ducks clinched Game Four to level the best-of-seven series at two games each.
The road team has won all four games between the cross-town rivals as the series heads back to Anaheim for Game Five on Monday.
Gibson was a surprise starter after backup goaltender Jonas Hiller finished Anaheim's Game Three victory.
Ducks coach Bruce Boudreau went with the hunch that Gibson would deliver in a must-win situation, and it paid off.
"As soon as we called Gibson up, I thought about starting him, and the overriding reason was I thought... he would give us the best chance to win," Boudreau said.
"When it comes down to it, in each individual game, like I've said before, it's its own entity, and whatever line up I put in, it's those guys."
The Ducks opened the scoring on Smith-Pelly's third goal of the season at 16:02. Getzlaf's shot from the slot went wide of the Los Angeles net, winger Corey Perry gained possession behind the net and fed Smith-Pelly with a no-look pass for a short shot that beat Los Angeles goalie Jonathan Quick.
Anaheim doubled its lead when Getzlaf scored his fourth goal of the playoffs on the power play at 18:45 of the first period.
With the Ducks up by two after the first period, Kings coach Darryl Sutter sought to change the momentum by pulling Quick in favor of rookie goaltender Martin Jones.
Quick could not be faulted on either Anaheim goal and stopped nine of 11 shots through 20 minutes.
The strategy apparently motivated the Kings as they outshot the Ducks 12-0 in the second period, but they could not dent Gibson. It was the first time in Anaheim playoff history that it failed to get a shot on goal in a period.
Jones made three saves in two periods.
"I don't think goaltending is an issue," Sutter said. "It's not shots; it's the ability to finish it off. We have trouble scoring against Anaheim; they're doing a good job at keeping us on the outside."
(Editing by Andrew Both/Amlan Chakraborty)