Martin Havlat and Thornton also scored for the Sharks, who won their second straight. The Sharks have three consecutive home victories for the first time this season.
"You have to take care of your home ice," Thornton said. "Now we need to build on this and start a little streak going."
"It came down to special teams, and it was tough to kill off so many," Oilers' captain Shawn Horcoff said. "You can't play hockey like that."
Havlat and Thornton each ended long goal-scoring droughts, and Marleau netted his second in three games.
Eberle recorded a point for the eighth time in nine games. Jones scored for the first time since he had a hat trick on Dec. 2.
"There were a lot of positive things," Sharks coach Todd McLellan said. "We overcame a sloppy power play and we won the third period."
Havlat scored midway through the first period to give the Sharks a 1-0 advantage. He played give-and-go with Michal Handzus and had an easy chance, scoring his first even strength goal of the season and his second goal. He had not scored a goal in his previous 17 games. Dan Boyle picked up a loose puck to create the scoring opportunity.
"Everybody knows our power play has struggled of late," Boyle said. "And then we get the game-winning goal on one with guys working hard on a puck behind the net."
Jones took advantage of a miscue in the neutral zone to score a short-handed goal and give the Oilers a 2-1 edge early in the second period. Havlat missed a pass and was beaten to the puck by Jones.
"We played a very simple game, and we still allowed too many shots," Jones said. "I don't know the stats but we must be one of the top five teams in the league in taking penalties."
Thornton scored his first goal in 14 games, and the Sharks scored their first 4-on-4 goal of the season to tie the game.
Joe Pavelski picked up a puck behind the net and started a wraparound, getting the puck to Marleau for the game winner.
"We're trying to simplify things on the power play and get shots through," Marleau said. "Pav made a great play to get it to me."
The Oilers are 2-7 in their past nine games. ... Thornton has 36 points in 34 games against the Oilers. ... Havlat left the game in the third period after hurting his ankle trying to get on the ice. ... Sharks D Colin White was taken off the ice with an apparent hand injury. ... The Sharks were 2-of-35 on power plays before Marleau scored. ... Jones scored his first goal in nine games against the Sharks. ... The Sharks are 41-6-1 in games in which Marleau and Thornton scored goals, dating to Thornton's debut with San Jose midway through the 2005-06 season. ... Sharks D Matt Irwin was reassigned to Worcester of the AHL after the game.