Dan Boyle also scored and Joe Thornton had two assists as San Jose, 5-1-1 in its last seven games, reached 50 wins for the second time in franchise history.
"We have a lot of new faces in here but what can you say, the depth of this organization is pretty strong," Boyle said.
The Sharks also moved four points ahead of Detroit in the race for the Presidents' Trophy.
San Jose used four forwards who were playing with Worcester of the AHL less than four weeks ago. Captain Patrick Marleau was the latest to join the list of injured players, sitting out due to a lower body injury suffered Saturday night against Phoenix. Marleau is not expected to play Thursday night in Edmonton, either.
"Anytime you don't have nine players, you want the other guys to step up and play well," Nabokov said. "Our system has changed a little bit, we're not as aggressive and we have to be a little bit smarter, and in a way, I think it's helped us."
Jarome Iginla got his 32nd goal of the season for Calgary late in the first period.
The Flames have lost eight of their last 12 as their lead atop the Northwest Division has been cut to just one point over Vancouver. The Canucks have seven games remaining and Calgary has six.
"I've always told our club and any club that I've ever coached, if you're playing against a team with a lot of injuries, often it's a very dangerous team because you have a lot of people, particularly this time of year, that are receiving more ice time than they ever expected," Flames coach Mike Keenan said.
San Jose took advantage of an early two-man advantage after Adrian Aucoin's clearing attempt went over the glass for an automatic delay of game penalty. Just 35 seconds later, Thornton found Boyle with a centering pass and Boyle fired a slap shot into the top corner to put the Sharks up 1-0 at 6:11 of the first period.
Iginla tied it with 4:51 left in the period, taking a pass from Daymond Langkow at the face-off dot and snapping a shot that Nabokov only got a portion of.
Cheechoo, back in the Sharks' injury-riddled lineup after missing two games with a lower body injury, then deflected Brad Lukowich's slap shot from the blue line past Miikka Kiprusoff with 7:36 left in the second with the game-winner.
Nabokov made the lead stand with 12 saves in the third period, including several clutch stops. His best save came on a power play midway through the period when, while sprawled on the ice, he reached back and was able to smother a shot from Mike Cammalleri.
"I tried to wait him out. I tried to get it back. He made a good play to get his glove back there," Cammalleri said. "Hindsight's 20-20. Instead of jamming it, I should have just flicked it quicker."
Earlier in the period, Nabokov also made an alert glove save off Eric Nystrom and kicked out a pad to stop Adam Pardy from in-close.
San Jose LW Milan Michalek snapped a six-game points streak (three goals, four assists). ... San Jose has killed off 17-of-18 penalties over the last seven games. ... Calgary lost D Cory Sarich late in the third period when he took a slap shot from Thornton off his skate. He was noticeably favoring his right foot as he was helped off the ice and went straight to the Flames' dressing room.
Copyright by STATS LLC and The Associated Press
Arena: Scotiabank Saddledome Location: Calgary, Alberta
Referees: Eric Furlatt, Brad Meier Linesmen: Don Henderson, Brad Lazarowich
Attendance: 19,289 (100.0% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Dan Boyle (15)
(Power Play) Assists: Joe Thornton, Joe Pavelski
Jarome Iginla (32)
Assists: Daymond Langkow, Dion Phaneuf
Jonathan Cheechoo (10)
Assists: Brad Lukowich, Joe Thornton