The trade was finalized after Kelley cleared waivers and the Nationals were free to deal him to any of the other 29 clubs. Baseball's non-waiver trade deadline was Tuesday, and all players have to be exposed to the waiver process in August before being dealt.
Washington's return for Kelley was international slot money. Washington also sent cash to Oakland. To clear a spot on their 40-man roster, the A's placed outfielder Jake Smolinski on the 60-day disabled list with a blood clot in his left calf.
Kelley, 34, is 26-23 with a 3.73 ERA in 10 MLB seasons as a reliever with the Mariners, Yankees, Padres and Nationals. He signed a three-year, $15 million deal with Washington in December 2015.
The Nationals designated Kelley for assignment after he threw his glove to the mound in anger late in a 25-4 victory over the New York Mets on Tuesday. Kelley said he was frustrated with the umpiring crew and subsequently apologized, but Nationals general manager Mike Rizzo called it a "selfish act," and manager Dave Martinez said it was "disrespectful to the organization."
Kelley adds another arm to an Oakland bullpen that ranks fifth in MLB with a 3.34 ERA. Oakland's relief contingent is led by All-Star closer Blake Treinen, who has 28 saves and a 0.95 ERA, and setup man Lou Trivino, who has a 0.94 WHIP and a 1.18 ERA.