Rookie Tomas Hertl tied it with 10:43 left in regulation for the Sharks, who survived a much-improved effort from their bitter California rivals after two blowout wins in San Jose.
Marleau ended it for the Sharks with his third goal of the series. His shot banked off Kings defenseman Slava Voynov's stick on its way past Jonathan Quick, who made 36 saves in his third straight loss.
Antti Niemi stopped 28 shots for the Sharks.
Game 4 is Thursday in Los Angeles.
The Sharks have won five straight overtime playoff games and 10 of their past 11, with Marleau scoring four game-winning goals during that remarkable stretch.
Los Angeles has slipped to the brink of first-round elimination just two years after its Stanley Cup title run. The Kings were the NHL's best defensive team during the regular season, but they've allowed 17 goals in the first three games of this series.
Only three teams have ever rallied to win a best-of-seven NHL playoff series after trailing 0-3.
Jeff Carter tipped in a tiebreaking goal for Los Angeles on a power play early in the third period, but Tomas Hertl evened it right after a power play expired with 10:43 left. San Jose dominated the third, but Quick made 23 saves to send the Kings into their first overtime playoff game at home in three years.
Matt Nieto got his first career playoff goal for the Sharks, and Brent Burns also scored.
Marian Gaborik scored and Jarret Stoll ended his 29-game playoff goal drought for the Kings, who stumbled back home after opening the series with two disastrous games at the Shark Tank.
The Kings' nervous fans got quiet early on when Burns partly whiffed on a wrist shot and produced a knuckling puck that sailed past Quick just 10 seconds into a Sharks power play.
Stoll evened it with his first playoff goal in two years early in the second period. Stoll hadn't found the net in the postseason since his series-clinching overtime goal against Vancouver in 2012.
Gaborik then put the Kings ahead all by himself, lugging the puck from the opposite blue line and beating Niemi with a vicious backhand.
San Jose evened it 1:18 later when Nieto, from nearby Long Beach, scored into an open net. Quick had been knocked to the ice by Kings defenseman Robyn Regehr moments earlier.
Carter capitalized in the waning seconds of a power play by tipping Anze Kopitar's shot in front for his first playoff score since Game 2 of last season's Western Conference finals, ending a personal 393:34 drought.
Hertl converted his own rebound midway through the period to even it again, scoring on the rink where a knee-on-knee hit from Dustin Brown sidelined the Czech rookie for 45 games earlier this season.
NOTES: The Kings scratched F Colin Fraser after recalling the two-time Stanley Cup champion from the AHL earlier in the day. Los Angeles also scratched F Kyle Clifford, who went scoreless with a minus-3 and 16 penalty minutes in the first two games, and D Matt Greene, a minus-4 in the series. ... The Kings dressed F Tanner Pearson, who made his NHL playoff debut last season before he had ever played in a regular-season NHL game. ... Los Angeles played its 41st playoff game in the past three seasons, the most by any NHL team in that span.