Burns was called for high sticking, and San Jose was assessed a minor penalty for delay of game for the unsuccessful challenge.
Jeff Carter scored on the ensuing 5-on-3 power play at 6:56 to make it 6-2.
"We've blown a lot of third-period leads this year," Doughty said. "If you want to be a good team in the NHL, you have to learn how to seal the deal when you have the lead and finish the game off."
Jones, making his third straight start, was pulled after Carter's goal. Sharks coach Bob Boughner indicated that Jones might have been removed earlier if Devan Dubnyk was the backup.
"That's not up to our standard, that goaltending," Boughner said. "We gave him a chance to grab the net, especially with [Dubnyk] being hurt. We're not expecting him to win hockey games for us standing on his head. But we've got to have solid goaltending, especially at the beginning of games. It's just not up to our standard, and I'm sure it's not up to his own."
Dubnyk is day to day because of a lower-body injury, the Sharks said earlier Thursday.