Five assists and goal for Messi, hat-trick for Suarez as Miami hit six
Two-goal substitute Matias Rojas sparked the second half rout from Eastern Conference leaders Inter after New York had led 1-0 at the break.
Without Messi, Miami had fallen to a 4-0 defeat to the Red Bulls in New York in March, but they more than avenged that loss with a merciless second-half demolition job.
Messi's five assists, which all came in the second half, are a new record for an MLS game and he now has 10 goals and 12 assists in eight league appearances so far this season.
Operating behind Suarez, Messi's passing through the slightest of gaps, taking advantage of his former Barcelona teammate's telepathic understanding of his intentions, was simply too much for the Red Bulls, who came into the game third in the East.
"Obviously what we have seen today surprises people, because you don't see that in football," said Suarez of Messi's five assists.
"But as a teammate and someone who knows him, nothing surprises me about him," added the Uruguayan.
The 37-year-old Suarez's triples takes his own tally to 10 in 11 league games for Miami and despite no longer being blessed with pace, his deadly finishing clearly remains intact.
While it was the kind of attacking display that Miami owners Jorge Mas and David Beckham surely dreamed of when they brought in the star pairing, there was no sign of the explosiveness in the opening half.
Dante Vanzeir had put the Red Bulls ahead in the 30th minute, slotting home after Wikelman Carmona's shot came out off the post, following a superb through pass from Swedish playmaker Emil Forsberg.
Gerardo Martino's Inter lacked intensity and ideas in the opening 45 minutes but the coach's half-time change transformed the game.
Paraguayan Rojas was introduced for the second half and took just three minutes to make an impact, collecting a pass from Messi and evading two defenders before unleashing an unstoppable drive from 25 yards out.
Messi then put Miami in front when Rojas robbed Carmona and found Suarez, who slipped the ball to the Argentine, who made no mistake.
The eight-time Ballon d'Or winner then turned provider again as he split the New York defense with a brilliant through ball to Rojas, who deftly chipped over the advancing Carlos Coronel.
Suarez's hat trick then came in the space of 12 minutes -- he first found the target when Messi chipped in a ball from the right and the Uruguayan hooked home a volley.
It was playground stuff minutes later when Suarez played a one-two with Messi and then poked home to make it 5-1.
Messi was yet again the creator for Miami's sixth, which Suarez turned in from close range from the tightest of angles.
"The partnership of him (Messi) and Luis (Suarez) in the second half worked very well, like in old times (...) They were very influential in the game," said Martino.
New York had a late consolation in the final seconds with a penalty from Forsberg.