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.