The World Cup winner again turned provider as he split the New York defense and delivered a brilliant ball to Rojas, who cleverly beat the onrushing Carlos Coronel.
Suárez's hat trick came in the space of 12 minutes – he found the target first time when Messi fed a ball in from the right and the Uruguayan volleyed home.
A few minutes later it became a level playing field when Suarez played a one-two with Messi and then made the score 5-1.
Messi was once again the creator of Miami's sixth, which Suárez headed in from close range from the most difficult of angles.
New York, which came into the game ranked third in the East, got a consolation from Forsberg's penalty in the final seconds.