The five-time defending NL West champions have Kenley Jansen as their closer, but he's been shaky since coming off the disabled list, allowing seven earned runs in four innings over four appearances.
Madson was 5-5 with five saves and a 4.08 ERA in 69 games for Washington. The 38-year-old recently came off the DL after being sidelined by nerve irritation that caused pain in his neck and back.
''Feels like a fresh start, especially to a year I've struggled a little bit,'' Madson said as he was leaving Nationals Park. ''Now none of that matters. All that matters is from this point forward and helping that team win games. That's it.''
Madson, who helped Philadelphia and Kansas City win World Series titles, was acquired by the Nationals with fellow reliever Sean Doolittle from Oakland in July 2017.
''Just push the ball to Kenley and that sort of reminds me of old times with (Brad) Lidge. Just push the ball to Lidge and we win,'' Madson said of his time with the Phillies. ''So, push the ball to Kenley and we win.''
For his career, Madson is 61-48 with a 3.45 ERA and 91 saves.
''What he's done is proven, pitched in big spots,'' Dodgers manager Dave Roberts said. ''We're betting on the stuff and the person.''
After joining the Nationals last July, Madson went 3-0 with a 1.37 ERA and one save in 20 games as Washington captured its second straight NL East title.
''It was a huge jolt to my career last year, coming into the playoff push and kind of revitalizing ... actually showed myself what I can do,'' Madson said. ''My stuff got so much better and so I really appreciate them bringing me over here.''
Los Angeles (72-62) began the day two games behind Arizona in the NL West and 2+ games back in the wild-card chase.
Roberts said Madson would arrive in Los Angeles on Friday night and be activated on Saturday.
''Just to have another leverage arm is good for all of us,'' Roberts said. ''He's a guy who throughout his career has actually gotten lefties out better than righties, so to have him at any point we want to play him, that's a good thing.''
Washington, which unloaded second baseman Daniel Murphy and first baseman Matt Adams on Aug. 21, received right-handed pitcher Andrew Istler from the Dodgers. Istler, 25, went 4-4 with a 2.37 ERA in 41 games (one start) between three levels of Los Angeles' minor league system.
Additionally, the Nationals recalled left-handed pitcher Sammy Solis from Triple-A Syracuse.
To make room on the 40-man roster, the Dodgers designated left-handed pitcher Adam Liberatore for assignment.
Washington (67-67), the two-time defending NL East champion, entered Friday 7+ games behind first-place Atlanta and 7+ back in the wild card.
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