Mason made 24 saves, Kris Russell scored during a power play in the second period and Columbus defeated the Los Angeles Kings 2-0 on Monday night.
Mason believes he's growing with experience.
"Playing a lot helps because you get into a rhythm. Shutouts are nice, obviously, but what we're looking for is wins right now," the 20-year-old goalie said after logging his fourth career shutout.
The 21-year-old Quick also faced 24 shots, but couldn't stop Russell's rebound late in the second period.
The Blue Jackets' defenseman scored Columbus' first goal in three games against the Kings this season. Jason LaBarbera shut them out 3-0 on Dec. 6, and Quick blanked them by the same score last Tuesday.
Russell's goal ended a Columbus scoring drought against Los Angeles that spanned 186 minutes, 39 seconds, dating back to Rick Nash's power-play goal in the second period of a 3-2 shootout loss on Feb. 10, 2008, at Columbus.
Russell knocked in a rebound of his own shot at 18:36 of the second period for his second goal of the season while Michal Handzus was serving an interference penalty.
Nash slid a pass through Los Angeles defenders from the right boards, with the puck glancing off a stick of one of the Kings' defenders. Russell, just to the left of the crease, wristed a shot that bounced off Quick and back to him, and Russell poked the puck high over the sprawling goalie and into the net.
Nash scored his 16th goal into an empty net with 1:15 to play.
In a game mostly dominated by the defenses, Mason made several fine saves, including blocking and smothering the puck with a crowd of players in front during a Los Angeles power play in the second period.
"We had opportunities in the previous two games against them and we didn't capitalize on them," Mason said of the Kings' two earlier shutout victories. "We only got essentially one goal tonight, but it counted and the guys did the rest of the job in front of me."
Columbus coach Ken Hitchcock was pleased with the play of the goalie and the defensemen.
"The seven defensemen, plus Mason, we're trying to make ourselves a very hard group to play against back there. We're not the prettiest group or the fanciest, but we kind of maul you a little bit," Hitchcock said.
"That's the way we're built. They're able to keep the shots to the outside and they're confident that he can stop those outside shots."
Quick also played well. In the second period, he made an alert save on a fine play by the Blue Jackets' Andrew Murray, who fought his way up the right boards with the puck and whipped a pass to Jakub Voracek in the slot. But Quick was squared up and blocked Voracek's wrist shot.
"He's giving us a chance to win the game every game that he's played and that's all you can ask from any goaltender," Los Angeles coach Terry Murray said. "We have to score some goals for him, he's played with a lot of energy and a lot of composure and he's giving us a good look.
"It's nice to see that we have good young guys like this in the organization."
Quick was drafted in the third round in 2005. Mason went in the third round the following year.