Detroit acquired infielder Steve Lombardozzi, minor league lefty Robbie Ray and reliever Ian Krol for Fister, a 14-game winner for the AL Central champions.
Tigers president and general manager Dave Dombrowski said the trade was not done to save money in hopes of keeping Max Scherzer beyond next season.
"It gives us some flexibility for some other things we want to do," he said on a conference call.
Dombrowski said dealing Fister did not have anything to do with his hope to give Scherzer, the AL Cy Young Award winner, a new contract before he becomes a free agent following the 2014 season.
"I don't think that plays into it at all at this point," Dombrowski insisted.
Detroit has had an active offseason so far, and Dombrowski said he isn't done making moves, saying his top priority is to add a closer.
The Tigers traded slugging first baseman Prince Fielder and his $214 million contract two weeks ago to the Texas Rangers in part to potentially save more than $75 million, giving them even more financial flexibility. They got second baseman Ian Kinsler for Fielder in the swap of All-Stars.
They also allowed All-Star shortstop Jhonny Peralta to leave as a free agent for St. Louis after replacing him during last season with rookie Jose Iglesias just before Peralta was suspended for 50 games in the Biogenesis drug scandal.
"We're not cutting payroll whatsoever," Dombrowski said.
The Tigers did agree to a $1 million, one-year contract with utility player Don Kelly that avoids arbitration earlier Monday.
Kelly played six positions for the Tigers last season at all three spots in the outfield along with third, second and first base while hitting .222 with six homers and 23 RBIs.
Fister will likely replace Dan Haren in the Nationals' rotation and join Stephen Strasburg, Jordan Zimmermann and Gio Gonzalez.
The 29-year-old Fister was 14-9 with a 3.67 ERA last season and was 32-20 in two-plus seasons in Detroit, which acquired him in a trade from the Seattle Mariners. He struck out an AL-record nine batters in a row against the Kansas City Royals late in the 2012 season.
The 6-foot-8 Fister fared well in the postseason, even when he was hit in the head by a line drive. He shook off the scary moment in Game 2 of the 2012 World Series against San Francisco and took a shutout bid into the seventh inning in what ended up being a 2-0 loss to the Giants. Over three postseasons with the Tigers, he had a 2.98 ERA in seven starts and a relief appearance.
"He's been a very solid pitcher for us since the day we got him," Dombrowski said. "He has pitched well in big games."
Dombrowski said he's not surprised at least some Tigers fans weren't happy when they trade was announced because the team traded the "known for the unknown."
"People in general don't like those type of moves," he said.
The Tigers still have Scherzer, Justin Verlander and Anibal Sanchez atop their rotation.
"This is an exciting day for the Washington Nationals," general manager Mike Rizzo said in a statement. "We feel we've added a talented, young veteran to our starting pitching corps. Doug is battle-tested through playoff experiences, and the depth he brings to our staff is exceptional. We are thrilled to welcome him aboard."
The 22-year-old Ray was 6-3 with a 3.11 ERA for Class A Potomac and was 5-2 with a 3.72 ERA for Double-A Harrisburg, giving the Tigers a pitcher they think will start the season at Triple-A Toledo with the ability to possibly pitch in the majors if needed in 2014.
Krol made his major league debut in June for the Nationals after they acquired him from Oakland. The 22-year-old lefty was 2-1 with a 3.95 ERA in 32 games in Washington.
The 25-year-old Lombardozzi hit .259 with two homers and 22 RBIs in 118 games for the Nationals last season. The switch-hitting sparkplug played 48 games at second base, 23 in left field and four games at third base in 2013. He will replace Ramon Santiago on Detroit's roster next season.
"This was not an easy trade for (the Nationals) to make, either," Dombrowski said.
Washington went into the offseason hoping to add another starter it could count after having some uneven results from Haren last season and Edwin Jackson in 2012. Both of those right-handers were seen as short-term solutions and they ended up leaving as free agents after pitching for the Nationals for one season.
Haren essentially replaced Jackson. Fister now essentially replaces Haren, who recently signed with the Los Angeles Dodgers after going 10-14 with a 4.67 ERA for Washington.
Also, the Nationals and Ross Ohlendorf agreed to a $1.25 million, one-year contract. The right-hander went 4-1 with a 3.28 ERA in 16 games, seven starts, last season. The 30-year-old Princeton star can earn up to $3 million in bonuses tied to starts, innings pitched and relief appearances.
AP Sports Writer Howard Fendrich contributed to this report.
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