After a scrappy opening, the away side took control, forcing their opponents back and winning several free kicks in good positions.
Midway through the first half, their pressure paid off. Dax McCarty played the ball to Henry, who strode forward and fired the ball in from just outside the box.
The former Arsenal forward almost doubled his tally five minutes later but was denied by a good block.
It didn't matter, though, as from the resulting corner Olave found space in the middle of the area and placed the ball into the bottom corner of the net.
United continued to throw their bodies on the line in the second half.
Tim Cahill and Henry both had shots blocked on the line as New York looked to wrap up the win.
Henry again missed the chance to finish the game shortly afterwards, when his curling effort struck the post and bounced away.
That was the last real chance of the game and it was the away side who took all three points and moved up to third in the Eastern Conference.