Only nine minutes elapsed when Justin Meram opened the scoring for the visitors by firing home from 12 yards in what proved a fast-and-furious start to the contest.
Despite being humbled 5-2 by Orlando City last time out, there was little sign of scarring from the hosts, who moved 2-1 ahead when Bradley Wright-Phillips's header beat Clark from close range midway through the half.
Kei Kamara spurned a good chance to level up on the 25-minute mark, but the hosts survived to secure a slender half-time lead.
After the break, Clark pulled off two smart saves in quick succession to keep out Kljestan and Wright-Phillips just before the hour mark as New York hunted a two-goal cushion.
The Crew grew stronger as the half went on and they would have salvaged a point if not for Luis Robles, who pawed away Federico Higuain's fierce volley, but the Red Bulls held on.
Consequently, the hosts, who have four games to play, return to the summit of the Eastern Conference with an eight-point cushion from Montreal Impact in sixth.