Outfielder Matt den Dekker will head to Washington, while minor league right-hander Cory Mazzoni and a player to be named head to San Diego.
Blevins, 31, went 2-3 with a 4.87 ERA in 64 relief appearances for the Nationals last season. Left-handed batters hit .160 against him.
Torres, 27, went 2-1 with a 3.33 ERA in 70 relief appearances last season. He has averaged 5.5 walks per nine innings during his major league career.
Mets general manager Sandy Alderson said Blevins will serve as a lefty specialist, while Torres will more often be used for full innings.
"What was something of a glaring weakness I think now becomes -- could become -- a strength," Alderson said.
Alderson estimated the Mets spoke with six or seven teams about acquiring left-handed relief pitching in recent days.
Den Dekker helps fill a temporary void for the Nationals while outfielders Jayson Werth and Denard Span deal with injuries. Den Dekker was not slated to make the Mets' Opening Day roster because fellow lefty-hitting backup outfielder Kirk Nieuwenhuis was out of options, whereas den Dekker has an option remaining.
Torres became the first player in Major League Baseball to wear an isoBlox protective cap last season and said he was motivated to do so from watching former Tampa Bay Rays teammate Alex Cobb suffer a slight concussion after being hit by a line drive in a game in 2013.
Blevins will earn $2.4 million this season. He then is eligible for free agency. Torres will make $522,900. He is not eligible for arbitration until after the 2016 season and is under the Mets' control for five full seasons.
The Mets needed lefty relief help once Josh Edgin underwent Tommy John surgery two weeks ago.
Left-hander Sean Gilmartin, a Rule 5 pick from the Minnesota Twins, is likely to make the Opening Day roster even with the acquisitions of Torres and Blevins, manager Terry Collins and Alderson suggested.
Mazzoni, a 2011 second-round pick from NC State, missed the first two months of last season with a side-muscle injury. He went 5-1 with a 4.67 ERA in nine second-half starts for Triple-A Las Vegas after healing.