Hudson returned to the Nationals on a two-year contract Monday, Jeff Passan of ESPN.com reports.
Hudson earned the save in Tuesday's 7-4 win over the Cardinals in Game 4 of the NLCS, giving up one walk in 1.1 innings.
Hudson was reinstated from the paternity list and added to the Nationals' NLCS roster Saturday, Jesse Dougherty of The Washington Post reports.
Hudson is on paternity leave and will miss the first game of the NLCS against the Cardinals on Friday, Mark Zuckerman of MASN Sports reports.
Hudson earned the save against the Phillies on Thursday after tossing a scoreless ninth inning, allowing one walk and striking out three.
Hudson pitched a perfect ninth inning to record his sixth save of the season in Tuesday's 4-1 win over the Phillies in the first game of a doubleheader.
Hudson gave up three hits over two scoreless innings to record his fifth save of the season in Friday's 6-4 win over the Marlins.
Hudson gave up a hit and struck out two over two scoreless innings Tuesday to record his fourth save of the season in a 6-2 win over the Cardinals.
Hudson retired two batters and served up three unearned runs on two hits and a walk in Tuesday's 11-10 win over the Mets.
Hudson will be part of a closer-by-committee situation while Sean Doolittle (knee) is on the 10-day injured list, Jamal Collier of MLB.com reports.