Known more for his fighting than his offensive ability, Brashear provided the game's first goal with 4:42 left in the second period, beating Martin Biron with a wrist shot from the top of the left faceoff circle.
It was the tough guy's second goal of the season and just his 58th in 550 career games.
With Roman Cechmanek getting the night off, Esche had little trouble with one of the NHL's worst offenses, making 10 saves in the first two periods before turning aside five shots in the third en route to his fourth career shutout.
Defenseman Eric Desjardins added an insurance goal with 5:55 remaining for Philadelphia, which extended its winning streak to three games.
Buffalo was blanked for the fourth time this season and has been held to two goals or fewer in 19 of 29 games.
Celebrity Families of 2014 [PHOTOS]