Calgary made it 2-1 at 12:17 on Stone's slap shot from the right point.
"They won a little bit of the war of the paint, the blue paint ," Flames coach Glen Gulutzan said. "We were around it, we put a lot of rubber there, we were in front of him but we just couldn't get anything to go."
Jones made 27 saves in the final two periods.
"I thought Jonesy was our best player," DeBoer said. "I thought he was outstanding. Otherwise it could have been a 5-4 or 5-5 game."
Kane, who scored an NHL career high four goals against Calgary on March 16, scored a shorthanded goal at 8:26 of the second period to give San Jose a 3-1 lead.
Justin Braun scored from the right circle at 10:09 of the third period to make it 4-1, and Kane scored at 12:10 when Joe Pavelski tipped Dillon's shot and the puck deflected off Kane for the 5-1 final.