In return for the veteran forward, the Flames received second- and third-round draft picks in 2015.
"Curtis is a player we coveted and we are pleased that he decided to waive his no-trade clause to join our team," Capitals general manager Brian MacLellan said in a statement. "We felt we needed to get deeper and more experienced at the forward position. Curtis plays important minutes and his presence on-and-off the ice will benefit our young forwards."
Glencross is set to become an unrestricted free agent after this season, his seventh with the Flames. The 32-year-old Saskatchewan native has nine goals in 53 games for the Flames this season.
"As tough as it is to say good bye to everyone in Calgary and the fans -- I've had such great support over the years -- I'm excited for a new start," Glencross said.
Glencross will provide size and a physical presence for the Capitals as well as a body in front of the net capable of playing in Washington's top six.
Glencross has a salary cap hit of $2.5 million, and the Capitals will assume his entire salary.
The NHL trade deadline is 3 p.m. ET on Monday.
Information from The Associated Press was used in this report.