The Sharks get a 2020 second-round draft pick (originally from Colorado) and a conditional 2021 third-round pick. San Jose retains half of Dillon's $3.27 million cap hit.
"Brenden is an experienced defenseman who plays a solid defensive game with a high compete level and physicality," Capitals general manager Brian MacLellan said in a statement. "We felt it was important for us to add a player of his caliber to our defensive group."
Dillon is set to be an unrestricted free agent this offseason after completing a five-year contract. That deal came after he joined the Sharks in an in-season trade from the Dallas Stars.
Dillon, 29, has one goal and 13 assists in 59 games. He ranks fifth in the NHL this season with 83 penalty minutes and has 62 games of postseason experience, with 60 of those coming with the Sharks, including their run to the Stanley Cup Final in 2016.
The Sharks, at 26-29-4, are likely to see a four-year playoff streak end this season. San Jose is expected to continue to sell ahead of the NHL's Feb. 24 trade deadline.
"Brenden Dillon has been a leader on and off the ice for our team for the past four-plus seasons," Sharks GM Doug Wilson said. "Few players in Sharks history have displayed the class and character that Brenden exudes. ... However, because of where we find ourselves in the standings this season, it's imperative that we look towards acquiring assets for the future."
The Capitals have had the best record in the league for chunks of this season, but they are 4-6 in their past 10 games. Pittsburgh is only one point behind in the tough Metro Division with two games in hand.
The Associated Press contributed to this report.