The shortstop selected 13th overall by San Diego in the June 2014 draft was part of a three-team trade in December that sent 2013 AL Rookie of the Year Wil Myers to the Padres.
Nationals prospects Steven Souza and Travis Ott went to Tampa Bay, and Washington acquired right-hander Joe Ross and a player to be named.
Turner hit .322 at Double-A for the Padres organization. The Nationals plan to assign Turner to Double-A Harrisburg.
''From the outset, this was a complex deal, but we thought it was one that would better our organization for the present as well as the long term,'' Nationals general manager Mike Rizzo said.
Ross threw eight solid innings to pick up his first career win in the majors on Saturday when Washington beat the Milwaukee Brewers 7-2.