The best-of-7 series is tied 2-2. Game 5 will be in Edmonton on Tuesday.
Quick broke a tie with Mike Richter for most shutouts in the playoffs by a United States-born goalie. It was his first since June 9, 2014, in Game 3 of that Cup Final against the New York Rangers.
"I mean, we played the game that got us to the playoffs," Quick said. "We got pucks deep, we forechecked, blocked shots, (got) pucks to the net and rebounds. We play that way, we can be successful."
The Kings, the No. 3 seed from the Pacific Division, were outscored 14-2 in the previous two games. They lost 8-2 at home in Game 3 on Friday.
"Obviously, I think everyone in our room was tired of giving up that many goals so, as a whole, I think we tried to lock it down more," Los Angeles defenseman Mikey Anderson said. "Get back to our identity, being harder to play against, and trying to give up less than we've given up the last couple games."
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Carl Grundstrom had two goals and an assist, and Trevor Moore and Troy Stecher each had a goal and an assist for the Kings, who had lost six straight home playoff games. They had not won here since defeating the Rangers in Game 5 of the 2014 Cup Final to win their second championship.
Mike Smith made 42 saves for the Oilers, the No. 2 seed from the Pacific.
"I think it's the same thing across the board, probably," Edmonton forward Ryan Nugent-Hopkins said. "Penalty kill, five-on-five, power play, it's just working. We got to outwork those guys."