Abad, who will turn 27 in December, pitched to a 3.35 ERA in 37 2/3 innings for Washington this year, averaging 7.6 strikeouts and 2.4 walks per nine innings. He was taken off the Nationals' 40-man roster last week when he was designated for assignment.
Abad spent 2010-12 with the Astros and has a career record of 1-14 with a 4.56 ERA.
The 22-year-old Wooten, a 37th-round Draft pick by the A's in 2012, hit .257 with 20 home runs in 133 games for Class A Beloit this year.
This marks the fifth trade consummated by the A's and Nationals in the last three years.
Jane Lee is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.