Adam Fox also scored and Igor Shesterkin had 21 saves to help the Rangers win for the third time in five games (3-1-1).
''Obviously it was an important goal today,'' Gauthier said. ''It was nice to get it going. ... We're building some good chemistry and it was just nice to get rewarded.''
Logan Couture scored for San Jose with 17 seconds left to break up Shesterkin's shutout bid. James Reimer stopped 41 shots.
''It was a step back tonight,'' said Sharks coach David Quinn, who was fired by the Rangers after the 2020-21 season. ''(Over) 82 games, you're gonna have nights off. `Okay, let's see how we respond.' That's the key. ... But we stunk. ... It hasn't happened in a long time. So, let's see how we respond on Monday.''
Gauthier got the Rangers on the scoreboard with 6:03 left as he tipped Jacob Trouba's hard shot from the point past Reimer for his third of the season.
Fox had an empty-netter with 1:07 left to put the Rangers up 2-0. It was his sixth.
''We were a little frustrated, the puck didn't go in the net. Trust me, but they didn't lose their focus,'' Rangers coach Gerard Gallant said. ''They kept pushing and pushing because they were playing well, and the chances were there. There were a lot of Grade A chances and Reimer made some great saves.''
Couture, standing on the right doorstep, then got a pass from Tomas Hertl from in front and quickly put it past Shesterkin for his his ninth.
''That's a really good hockey team over there and we stayed right with them,'' Reimer said.
Sharks forward Nico Sturm went to the locker room after getting boarded by Rangers defenseman Ryan Carpenter late in the first period but returned to the game in the second. Carpenter was issued a two-minute penalty for boarding.
Rangers F Barclay Goodrow played at SAP Center for the first time since San Jose traded him to Tampa Bay on Feb. 24, 2020. ... Sharks D Mario Ferraro appeared in his 200th NHL game.
Rangers: At Los Angeles on Tuesday night.
Sharks: Host Ottawa on Monday night.
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