The 37-year-old Thornton went 0-3 with a 2.55 ERA in 24 2-3 innings across 46 appearances with the Yankees this season.
"He gives us another power option out of the bullpen," Nationals general manager Mike Rizzo said before his team played the New York Mets. "He's a hard thrower that's had great success against lefties and righties in his career."
Both Rizzo and manager Matt Williams spoke highly of Thornton's postseason experience, with the GM going so far as to say Thornton "has pitched in the World Series." But Thornton's playoff record consists of three appearances covering 3 1-3 scoreless innings for the White Sox during the 2008 AL division series against Tampa Bay.
Thornton was with the Boston Red Sox last season but was left off their postseason roster.
"He's just got experience. He's been there, he's done it," Williams said. "He's had times where he's come in late in games during pennant stretches."
Nationals relievers, including NL All-Star Tyler Clippard, have been one of the team's strengths most of the season, but recently they've tailed off, allowing 19 runs in 33 1-3 innings entering Tuesday.
Thornton was traveling to Washington and was not available for Tuesday's game.
In other moves, the Nationals transferred right-hander Taylor Jordan to the 60-day disabled list, making room on the 40-man roster for Thornton.
They also put catcher Wilson Ramos on the paternity list and recalled catcher Sandy Leon from Triple-A Syracuse. Ramos' wife gave birth to their first child Tuesday, Williams said.
Ramos is hitting .290 with three homers and 25 RBIs in 49 games this season. Leon rejoins the Nationals for the third time this season.
Also Tuesday, Rizzo said third baseman Ryan Zimmerman has a grade-three strain to his right hamstring and he is expected to be out about six weeks. Zimmerman was injured July 22.
Asked whether Zimmerman will return before the regular season ends, Williams replied: "I believe so." He said Zimmerman walked on a water treadmill Monday as part of his rehab work.