Boyd Kane had the other goal for the Flyers (3-3).
Brian Sutherby scored for the Capitals (2-2-1), who failed on seven power play opportunities including a two-man advantage in the last 57 seconds.
Eager flipped a shot from between the circles past Washington goaltender Brent Johnson to break a 1-1 tie.
Kane deflected a shot by Braydon Coburn into the net to put the Flyers ahead 3:56 into the first period.
Sutherby pulled the Capitals even at 13:22 of the period when his shot rolled past the outstretched glove of Flyers goaltender Martin Biron.
Philadelphia was without four players who were injured in the Flyers' 4-2 loss at Ottawa on Tuesday night.