The storm, named St Jude after the patron saint of lost causes, brought down trees and power lines across the southern UK, cutting off power to more than 200,000 customers.
EDF Energy said both units shut down safely and back-up diesel generators had kicked in to run safety systems.
Martin Pearson, station director at Dungeness B, said: “This is a scenario we are well prepared for and we quickly responded calmly and professionally to the loss of supply.
“The reactors are safely shutdown and National Grid staff are now working to restore the supply and once that is done we’ll bring both units back on line.”