Fleischmann's second goal this post-season snapped a 2-2 tie at 1:46 of the third following assists by Nicklas Backstrom and Alexander Semin.
Dave Steckel and Alexander Ovechkin (power play) scored first-period goals for the Capitals, who were playing a second-round contest for the first time since 1998.
Mark Eaton also scored once, while goaltender Marc-Andre Fleury was beaten three times on 26 shots for the Penguins, who pressured Varlomov with 11 of the game's first 13 shots, but scored just once.
Game 2 is Monday night in Washington.
Former Titans kicker dies in car crash
UPI Sports Results for September 20, 2014