The Caps also sent back a fourth-round 2015 draft pick in the deal. That pick was previously acquired by Washington from the Arizona Coyotes.
In making this swap, the Capitals add the 32-year-old Gleason, who has seven points in 55 games for the Hurricanes this season.
"We are pleased to welcome Tim to our organization," Capitals general manager Brian MacLellan said in a team-issued press release Saturday morning. "Tim is a veteran, physical defenseman and we feel his experience, toughness and skill set further strengthens our blue line."
Hillen, 29, has five points in 35 games for the Capitals this season.