PSG host relegation battling Le Havre on Saturday
Barcelona missed two penalties in normal time, with midfielder Miralem Pjanic being thwarted late in the first half by Cornella keeper Ramon Juan before Dembele hit straight at the feet of the 21-year-old in the 80th minute from yet another spot kick.
Dembele made amends with a vicious long-range strike in the second minute of extra-time which Ramon Juan got a hand to but was powerless to prevent it rocketing into the net.
Danish forward Braithwaite then sealed a nervy Barca victory moments after Cornella's Albert Estelles had been sent off.