Phil Kessel and Johan Larsson scored, and Darcy Kuemper made 27 saves for the Coyotes (21-24-5), who lost their ninth in 11 games.
"Obviously things haven't been going great for us," Kuemper said.
Arizona is in fifth in the eight-team West, three points behind the fourth-place St. Louis Blues for the final Stanley Cup Playoff berth from the division having played three more games. St. Louis defeated the Minnesota Wild 4-3 on Wednesday. San Jose is sixth, five points behind St. Louis, having played two more games.
The Vegas Golden Knights, Colorado Avalanche and Wild have qualified for the playoffs. The Golden Knights are six points ahead of the Avalanche and seven points ahead of the Wild for first place.
The Coyotes led the Blues by five points April 5.
"We're definitely going to fight right to the end," Kuemper said. "We're behind the eight ball here, but our efforts are going to be the same and we still believe. Obviously we've got to win a lot of games."
Evander Kane, Rudolfs Balcers and Kevin Labanc scored in the first period, and Josef Korenar made 28 saves for the Sharks (20-24-5), who swept the two-game set after losing their previous eight games (0-7-1).
"We got out to a lead tonight again, it always helps when you're playing from in front but, unfortunately, we had a lull where we let them back in it," Sharks forward Logan Couture said. "We really had to battle it out for the last 40 minutes."
The Sharks took a 3-0 lead, just as they did Monday before holding on for a 6-4 victory when Korenar made 21 saves for his first NHL win.
"Obviously a good start for us, and then they pushed back," Sharks coach Bob Boughner said. "I thought [Korenar] looked a little more comfortable in there. I know he wasn't happy about a couple of goals, but he went back and found his game."
Forward Patrick Marleau played his 1,600th game for the Sharks; only Gordie Howe (1,687 games for the Detroit Red Wings) has played in as many NHL games with the same team.
Kane scored for the third straight game, on a tic-tac-toe play following quick passes by Alexander Barabanov and Tomas Hertl for a 1-0 lead 1:12 into the first. Kane's 20th goal gives him six consecutive seasons with at least 20 and seven in his NHL career.
Balcers skated to a loose puck at the blue line and switched from his forehand to his backhand to score on a short breakaway to make it 2-0 at 10:06.