It turns out they both did it on the same night.
"The win is the most important thing, but it does feel nice to score," said Gomez, who has a goal and two assists in 15 games this season. "Obviously, it was time. I know I can score. It's just you've got to shoot to score, and sometimes I have trouble with that. I'm always looking to pass."
Meanwhile, Burish scored his first goal in 45 games on a short-handed breakaway to make it 2-0 San Jose early in the second period.
"If I start going out there thinking that I've got to score, I'm doing my linemates and my teammates a disservice," Burish said. "I've got to focus on other things: I've got to be solid defensively, I've got to kill penalties, I've got to bring energy. But if you can score goals, it feels a little better, too."
"Our record isn't as good of late, but we know we can play some better hockey," Canucks forward Mason Raymond said.
The Sharks started with seven straight victories, but their offense suddenly slowed. Heading into the game against Vancouver, they sat tied with Nashville for the fewest goals in the NHL with 47.
Although the Sharks didn't exactly overwhelm Vancouver with offense, they came on strong in the third period and took seven shots at Cory Schneider in overtime.
In the shootout, Logan Couture had to score in the third round or the Sharks would lose. Couture calmly deked to the backhand and beat Schneider between the legs.
After Niemi stopped Alex Burrows in the fourth round, Pavelski broke in and beat Schneider with a quick wrist shot to the blocker side to end it.
"You're always excited to win," Pavelski said. "It's not looking good there for a second, and then within a minute or two you've got a chance to win it. ... It's going to take more than two probably most nights, but if we can get two or three it's going to give us a good chance to win."
Chris Higgins had Vancouver's lone shootout tally, beating Niemi on the blocker side.
The Canucks fell behind 2-0 early in the second period, but it was their top line that got them back in it.
Daniel Sedin flipped the puck in front of the net and it took a fortuitous bounce to Henrik Sedin, who went to the backhand to beat Niemi at 13:23.
With 25 seconds remaining in the middle period, Hansen fired a quick wrist shot that Niemi never saw for his sixth goal of the season.
The Sharks opened up their lead thanks to Gomez and Burish.
Gomez took a soft slap shot from about 45 feet out that hit the crossbar and went in. Schneider appeared to be screened.
Gomez has struggled in recent years and was bought out of his contract by the Montreal Canadiens in the offseason. He was happy to sign a one-year deal with the Sharks for $700,000.
"I'm just enjoying each day now," said Gomez, who has 10 goals in his past 133 NHL games.
Niemi has been San Jose's bright light this season, and he continued his strong play. With Vancouver pressing for the winner late in the third period, he robbed Burrows and Henrik Sedin with nifty pad saves.
He had luck on his side, too, as Jason Garrison hit the post with a minute remaining.
In overtime, the Sharks killed off a two-minute Canucks power play with Gomez in the box for goaltender interference.
"What can you say about Nemo?" Gomez said. "He was definitely the story again tonight."
Vancouver, meanwhile, had plenty of chances to win, but couldn't convert on any of their five dangerous-looking power plays.
It's the sixth straight game the Canucks have failed to score with the man advantage.
"In the third period, they did everything they could but beat the goaltender," Canucks coach Alain Vigneault said of the power play. "The looks were there, the opportunities were there, their goalie made the saves."
Sharks coach Todd McLellan earned his 206th victory with the team, tying Ron Wilson for the franchise record. ... Stuart lost some teeth while killing a penalty in the third period, and blocked four shots. "He's black and blue with bruises," Burish said. "It looks like he went through a meat grinder." ... The Canucks went 0 for 5 on the man advantage, their sixth straight game without a power play goal. ... F Bracken Kearns, 31, played his first game as a Shark and the sixth of his career. Kearns is the son of former Canucks defenseman Dennis Kearns. Bracken Kearns, who was born in North Vancouver, had zero points in five games with Florida last season. ... Canucks D Keith Ballard was back in the lineup after being a healthy scratch the past two games. He left the game midway through the third period with a lower-body injury. ... Cam Barker also dressed for his second straight game, while Kevin Bieksa (groin) and Andrew Alberts (healthy scratch) did not dress. ... Martin Havlat (lower body), T.J. Galiardi (healthy scratch) and Justin Braun (healthy scratch) did not dress for San Jose.