Jackson (1-5), who went his first seven starts without recording a victory, finally earned his initial win as a Cub, limiting the Nationals to two runs on four hits and two walks in 5 1/3 innings.
The veteran right-hander struck out three.
His two-run double in the fifth inning broke a scoreless tie and he later scored on a two-run single by Anthony Rizzo (3-for-5).
David DeJesus also knocked in two runs for Chicago, which evened its three-game series.
Washington's Stephen Strasburg (1-5), meanwhile, was victimized for four unearned runs in loss, his fifth in seven starts since posting an Opening Day victory April 1.
The right-hander yielded five hits, walked two and fanned seven in five full innings.
Ian Desmond went 3-for-4 with a two-run homer for Washington.