LaRoche's RBI in first inning tied the game 1-1 and he scored on Mark DeRosa's two-run hit later in the frame, putting the Nationals ahead for good.
The Washington first baseman socked a solo homer in the third and drove in another run with a bases-loaded walk as the Nationals broke the game open with three more runs in the fourth.
De Rosa finished with three RBI as the National League East-leading Nats took the opener of a four-game series.
John Lannan (2-0) yielded three runs on four hits over six innings, overcoming five walks to earn the victory.
Tyler Clippard picked up his 22nd save with a 1-2-3 ninth.
Marlins starter Brad Hand (0-1) was roughed up in his first appearance of the season, allowing six walks in 3 2/3 innings. The left-hander was rocked for seven runs on six hits.
Emilio Bonifacio and Brett Hayes knocked in runs for Miami.
Teacher apologizes for showing sexual image of herself in class
Scarlett Johansson steps out with fiance after pregnancy reveal