Erat figures to provide scoring depth for the surging Capitals (17-17-2), who entered Wednesday just two points behind the Winnipeg Jets for first place in the Southeast Division.
Washington also received minor league forward Michael Latta in the deal.
The 31-year-old Erat, who has spent his entire 12-year career in Nashville, has four goals and 17 assists in 36 games this season.
"It is always difficult to trade a player who was drafted, developed and who has produced for the franchise for many years such as Martin Erat," Predators general manager David Poile said in a statement released by the team. "However, Marty came to us in recent days and indicated a trade might be in his best interests. Thus, the process began and we were able to explore options with our primary goal being to acquire a young top-end forward that would address our most critical need.
"We thank Martin for all his contributions to the Predators' success over the years and wish him and his family the best in Washington."
Forsberg was the 11th overall selection in the 2012 draft. The 18-year-old has spent this entire season playing in Sweden's second league.
"In Filip Forsberg, we are adding one of the top-rated young forwards in the world; we are excited to have him join our other emerging young talent and solid veterans at the position," Poile said. "We have used draft picks in recent years to add players at the trade deadline.
"Over the last several years, we traded away four first-round picks. It was imperative that we add a potentially dynamic offensive forward such as Forsberg. ... We have to be mindful of the long-term and this deal allows us to acquire a top-end young offensive forward that will help us."