Clinton-Dix is in the final year of his five-year rookie contract and will be a free agent after this season. He said earlier he probably would not be back with Green Bay next season, and he said on Monday he thought he might get traded.
Clinton-Dix tweeted a message to Packers fans on Tuesday that said, "Much love y'all" and included an image of a wedge of cheese.
He has never missed a game in his career and started 55 straight regular-season games in Green Bay. He has 378 tackles, 25 passes defended, 14 interceptions, 5.5 sacks and three forced fumbles in his career.
Clinton-Dix was a Pro Bowl selection in 2016, and he has three interceptions in seven games this season.
The trade strengthens a Redskins defense that was already strong. Washington ranks fourth in the NFL in total defense and fifth in scoring defense, yielding 19.1 points per game.
Clinton-Dix will pair with safety D.J. Swearinger Sr., who has four interceptions and two forced fumbles this season.
After acquiring Clinton-Dix, the Redskins waived linebacker Cassanova McKinzy and terminated defensive back Jason Thompson from the practice squad.