Chasetown made a lightning quick start and should have scored inside two minutes when Will Whieldon had an effort saved by Dan Haystead.
The hosts took the lead after 5 minutes. Liam Kirton got down the inside right channel and fed George Cater. His shot was saved but Ryan Wynter side footed home the rebound.
Whieldon went close again before the league leaders started to find a way back into the game. Tom Siddons glanced a header just wide from a Jon Challinor cross.
Cater was a whisker away from making it 2-0 at the start of the second half but Haystead held on.
Stamford thought they had equalised only for a goal line clearance to preserve Chasetown’s clean sheet.
Haystead then made a save high to his right to deny Kirton as the hosts looked to extend their lead.
Stamford began a spell of pressure and twice Curtis Pond had to save well to deny Ollie Brown-Hill and then Siddons.
The Lincolnshire side were awarded a spot kick in the 72nd minute after Mike O'Reilly handled on the line and Harry Vince converted to Pond’s left.
Stamford made their man advantage count as Siddons turned the ball home with barely a minute of normal time remaining.