After a physical first period featuring nine saves from Flyers netminder Michael Leighton, the Blackhawks broke through in the second frame with the pair of markers to take control.
Hossa collected a rebound of a Patrick Sharp shot and put it past Leighton 17:09 into the frame, and 28 seconds later Eager launched a long wrist shot from the right boards that sailed past Leighton's ear and into the net for a 2-0 Chicago edge.
Blackhawks goalie Antti Niemi turned aside 26-of-27 shots, allowing only Simon Gagne's score 5:20 into the third period. He was especially tough in the final frame, when the desperate Flyers launched 16 shots at him.
Leighton made 24 saves in Philadelphia's loss.
The Flyers will look to rebound when the series shifts to Philadelphia for Wednesday's Game 3.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys