Logan Couture tied it again 3-3 at 1:26 of the third, tipping Erik Karlsson's shot past Binnington.
"Loved the beginning of our third," Quinn said. "We come out, we get a goal, we hem them in, we had chances. The fourth goal [for St. Louis] is just one we can't give up."
Rosen gave the Flyers a 4-3 lead at 8:20 on a one-timer from above the left circle off a pass from Thomas.
"We played really well for spurts, but then just big-time mistakes ended up in our net," Couture said. "It's happening every single night. We've got to find a way to avoid making the big mistakes. Obviously, mistakes are going to happen throughout a game, but the big ones are killing us right now."
Noel Acciari scored an empty-net goal with 19 seconds remaining for the 5-3 final.
NOTES: Karlsson has 15 points during a six-game point streak (seven goals, eight assists). His 21 points in the first 15 games are tied for the most in San Jose history (Patrick Marleau in 2009-10). ... Meier has 10 points in a six-game point streak (six goals, four assists). ... Hertl has eight points in his past five games (two goals, six assists). ... St. Louis has won six straight against San Jose and is 7-0-1 in its past eight games against the Sharks. ... Blues forward Nikita Alexandrov played 9:31 and had two shots and two hits in his NHL debut.