With the game tied 2-2 and one out in the eighth, Jones hit a tiebreaking solo shot and McCutchen followed with another blast off Sean Burnett (3-5), handing the Pirates a win in the makeup of a game originally scheduled for May 17.
Jones finished 3-for-4 with a run-scoring single and Alex Presley chipped in with three hits in the Pirates' victory.
Jose Veras (2-2) earned the win with 1 2/3 innings of scoreless relief after starter James McDonald went 6 1/3 frames, yielding two runs on seven hits while striking out seven.
Joel Hanrahan collected his 24th save despite allowing a run in the ninth.
Nationals starter Livan Hernandez gave up a pair of runs in a seven-inning effort, yielding six hits and fanning six.
Rick Ankiel got two hits, including a solo homer, for Washington.