Hoops forward Scott Sinclair steered in the opener after 18 minutes to take his tally to the season to eight in the 2-0 win, with Dembele and Kieran Tierney hitting the woodwork before the break.
Celtic controlled the second half as well but could not add to their lead until Dembele chased what looked like a lost cause inside the Well penalty area, arriving at the ball before keeper Craig Samson who took his feet away.
The 20-year-old Frenchman picked himself up and slammed in the penalty for his 13th goal for Celtic since arriving from Fulham in the summer, sealing a win that kept the league leaders four points ahead of Aberdeen with a game in hand.
Rodgers said: "I said to the players before the game for us to succeed it's not just about us being hungry for the success, it's about being starving.
"You have to really, really push yourself in moments when you think things are gone and you have to fight. That (Dembele) was a perfect example of the spirit in the team in that they will chase everything.
"That looked like it was dead and was going to go through to the keeper but he made it something. The best strikers do that and create something out of nothing or make something for themselves.
"He made the penalty through his endeavour. Forget about his quality, that's about a mind-set to get there and he tucked it away really well."