The Nationals, who trailed 4-1 early in the game, chipped away at the Twins' lead with single runs in the third and fifth innings.
Washington tied it up in the sixth when Anthony Rendon singled off reliever Anthony Swarzak (1-2) and former Twin Denard Span delivered a two-out, run-scoring triple.
Then in the seventh, Adam LaRoche collected a one-out double and Desmond followed by belting a Swarzak offering into the left-center field gap, sending home the decisive run.
Jayson Werth provided two RBI for the Nats, who took two-of-three from Minnesota following a 7-0 victory in Sunday's first game.
Chris Herrmann went 3-for-4 with a two-run double and Pedro Florimon cracked a two-run homer for the Twins, who fell for the fourth time in five games overall.