Senators goalie Craig Anderson was injured during a scoreless overtime when Dallas' Valeri Nichuskin fell on top of him, delivering a thigh to his head. Anderson was taken off the ice on a stretcher and was replaced by Robin Lehner.
The contest moved to a shootout, where Benn beat Lehner with a shot to the glove side in the second round.
Kari Lehtonen then made a sliding stop on Ryan, and the Stars began a three-game road trip with a victory.
Nichuskin and Ray Whitney scored and Lehtonen finished with 33 saves for Dallas.
Ryan, Eric Karlsson and Jason Spezza each provided a goal an assist for the Senators, who fell in a shootout for a second straight game.
Anderson stopped 33-of-36 shot before leaving with his injury and Lehner turned aside both shots he faced in overtime.
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