On-loan forward Fabio assisted both goals for the Red Bulls (1-2-0, 3 points), who preserved their first shutout in five regular-season matches dating back to last year after conceding five goals in their first two games this campaign.
Clark, a 17-year-old who is reportedly heading to New York's sister club RB Leipzig in Germany following the season, departed later in the second half with an apparent minor knee injury.
Chicago (0-2-1, 1 point) rarely looked close to scoring and was outshot 14-6, including 6-4 in efforts on target.
Fire forward Robert Beric failed to score in a second consecutive match after a stretch of 11 goals in 14 league games dating back to last season, as his side remained winless on the road since the 2019 away finale.
After a tightly contested first half, the hosts took the lead two minutes into the second off a throw in. Fabio controlled it and worked into the penalty area against Francisco Calvo, then dragged a cross back toward the penalty spot. Casseres met it in full stride and hammered a finish past Bobby Shuttleworth.
New York doubled its lead 16 minutes later on another well-worked attack inside the box. Sean Davis lofted a pass toward Fabio just outside the 6-yard box. The Brazilian forward headed the ball down for Clark, who volleyed it with his first touch inside the right post.
The second goal prompted Fire manager Raphael Wicky to make a triple substitution, but the Red Bulls continued to control the game.
Daniel Royer twice missed on good opportunities in search of his first goal of the season. And after making his season debut as a substitute, Omir Fernandez hammered an attempt on a setup from Kyle Duncan that forced Shuttleworth into a good save.