Charlotte had won 11 of their last 13 regular-season games to secure home field advantage in the opening round as the fourth seeds.
But they were without a pair of top contributors with Spanish midfielder Pep Biel sidelined by injury and star forward Wilfried Zaha suspended after he was sent off late in Charlotte's victory over Philadelphia in the regular-season finale.
Martinez grabbed the lone goal in the 34th minute, gaining control of a headed pass by Andres Perea to turn and dribble through a clutch of defenders and unleash a right-footed shot from the center of the area into the bottom right corner.
NYCFC - one of the teams to beat Charlotte down the stretch - made the goal stand up to seize control in the best-of-three series.
The winner of the series will play either Philadelphia Union or Chicago Fire in the second round.
Top-seeded Philadelphia, winners of the Supporters Shield as the team with the best regular-season record in the league, beat Chicago in a penalty shootout in game one of their series.