Carlos Miguel Coronel made one save as the Red Bulls (7-11-5, 26 points) cruised to just their second win in their last 14 matches (2-7-5).
Nick Marsman finished with four saves for Inter Miami (9-10-5, 32 points), who saw their three-match winning streak and six-match unbeaten string (5-0-1) come to a halt. Miami remained one point behind seventh-place Atlanta United in the Eastern Conference.
Inter Miami played with 10 men after defender Jorge Nicolas Figal received a second yellow card in the 38th minute. Figal received his first yellow after a bad foul in the 10th minute before being whistled for an off-the-ball infraction at midfield.
New York nearly opened the scoring in the fourth minute as Fabio's shot eluded Marsman. The ball skipped toward the goal before being brushed aside following a diving clearance from Christian Makoun.
The Red Bulls made good on their next prime scoring opportunity in the 18th minute.
Wikelman Carmona deftly sent a lead pass to Klimala, who had gained a step on Inter Miami's Leandro Gonzalez Pirez. Klimala provided the finish by sending a left-footed shot past Marsman for his team-leading sixth goal of the season and second in as many matches.
New York doubled the advantage just 13 minutes later.
Thomas Edwards' throw-in bounced twice in the box before a charging Fernandez headed the ball home for his first goal of the season.
Fabio provided insurance after he headed home loose balls in the box in the 65th and 89th minutes. He has five goals on the season.