Leon Draisaitl had two goals and an assist, Bouchard had a goal and two assists, and Connor McDavid had three assists for the Oilers, who are the No. 2 seed from the Pacific and were trailing 3-0 after the first period.
"We are a very, very resilient group, and we're not going to quit ever, no matter what," Draisaitl said. "So, really proud of the guys, but that's only two (wins). It's first to four."
Jack Campbell made 27 saves after replacing Stuart Skinner, who allowed three goals on 11 shots, at the start of the second period.
"Not at easy situation," Draisaitl said of Campbell. "Hasn't been an easy year for him, but the way he showed up tonight was really, really great, and we're very fortunate to have him."
Viktor Arvidsson had a goal and an assist, and Kevin Fiala had two assists in his return from a lower-body injury for the Kings, who are the No. 3 seed from the Pacific. Korpisalo made 37 saves.
"We can be frustrated tonight, but tomorrow's a new day and we've got to get ready for Game 5," Los Angeles captain Anze Kopitar said. "The first period was obviously really good and the second period was not good, so there's no secret to that. We'll have to correct that going forward."
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Fiala had missed the final six games of the regular season and the first three of the series. He was originally injured on a knee-on-knee hit from Colorado Avalanche forward Andrew Cogliano on March 9.
"You can always improve," Fiala said. "I felt OK. Like you said before, I hadn't played in three weeks, so it's not going to be the best, but it was OK."