Cahill headed a rebound in the 80th minute after Andy Gruenebaum made the initial save off a shot by Jonathan Steele.
"Jonny hit a great strike," Cahill said. "The keeper reacted with a save. I seen the keeper off the line and just lifted it over with an instinctive header. Overall, it's a good reward for a hard day's work."
The Red Bulls (5-4-2) had three shots on goal, all in the second half, but still won their third straight overall. They improved to 3-0-2 against the Crew in the past five meetings.
Higuain took a stutter step before aiming his shot to the corner in a similar move to the one he successfully employed in the previous match against D.C. United on April 27.
"I unfortunately couldn't kick the way I'm used to," Higuain said. "I had the opportunity to convert that into a goal but just couldn't this time."
Robles, who had six saves, said watching video of the Crew definitely helped him prepare for Higuain's move.
"If I hadn't known that beforehand maybe I would have really oversold myself too quickly," Robles said.
Columbus (3-3-3) snapped its 12-game home unbeaten streak (8-0-4) and coach Robert Warzycha said the failed penalty kick was the difference.
"The penalty changed everything," he said. "If we would have scored the penalty we probably win the game."
The game's first shot on goal didn't come until the 50th minute when the Crew's Jairo Arrieta struck from 20 yards.