A Tadeo Allende hat-trick and goals from Mateo Silvetti and Telasco Segovia ensured Miami advanced to Major League Soccer's championship game next Saturday at their home field in Fort Lauderdale.
While Messi did not get on the scoresheet, the 38-year-old provided a crucial, classy assist for Miami's third goal scored by Silvetti that effectively killed off the game as New York chased an equalizer.
"Leo has accustomed us to the extraordinary," Mascherano said of his former Barcelona and Argentina team-mate.
"He's someone extraordinary, someone we'll never see again. Today, perhaps we're surprised that he didn't score, but he gave us peace of mind with the third goal, an assist that only he can see. He practically sealed the game."
Miami will host the Vancouver Whitecaps in next Saturday's MLS showpiece.
Messi and his team-mates will head into that game bristling with confidence after another ruthless, high-scoring demolition job that followed on from their 4-0 drubbing of FC Cincinnati in the previous round.
Allende - who scored twice in the victory over Cincinnati - was once again the focal point of the Miami attack.
The 26-year-old Argentine forward opened the scoring in the 14th minute after cleverly springing the New York offside trap.
A perfectly weighted ball from Sergio Busquets sent Allende racing in on goal, and the winger shrugged off Raul Gustavo's challenge to lash home and make it 1-0.
New York had barely had time to digest that early setback before it was 2-0.
Miami's former Spanish international full back Jordi Alba found himself in acres of space on the left flank and lasered a pinpoint cross into the New York area.
Allende darted into a gap and timed his jump to perfection as he glanced in a superb header that left New York goalkeeper Matthew Freese rooted to the spot as the ball flew into the bottom corner.
Miami's grip on the game was loosened though when New York scored against the run of play in the 37th minute, centre-back Justin Haak heading home from Maxi Moralez's free-kick to make it 2-1 at half-time.
New York grew in confidence and might have equalized in the 66th minute when Nicolas Fernandez picked out substitute Julian Fernandez inside the area.
Fernandez's curling shot was clawed away by Inter goalkeeper Rocco Rios Novo and the miss was to prove crucial.
Miami immediately swept upfield and a flowing move saw Rodrigo De Paul and Alba combine before Messi slipped a pass to Silvetti who rifled home to make it 3-1 after 67 minutes.
'Unbreakable'
Segovia then eliminated any doubt with Miami's fourth in the 83rd minute before Allende completed his hat-trick a minute from time to spark wild celebrations.
The victory leaves Miami just one win away from completing a remarkable transformation after a season that appeared to be unraveling in disarray earlier in the campaign following after a string of defeats.
"The real credit goes to the players who embraced where we wanted to go," Mascherano said. "We strengthened as a group and reached the end of the season with a brotherhood within the team, where everyone pulls together, and it doesn't matter who starts.
"We are a group, and the power of a group is unbreakable."
Meanwhile Vancouver - who thumped Miami 5-1 on aggregrate when they met in the CONCACAF Champions Cup semi-finals in April - signalled that they will be ready to take on the star-studded Florida outfit with a masterful display to beat San Diego.
The Canadian-based club got off to a dream start with Brian White firing them into the lead after only eight minutes.
More relentless Vancouver pressure forced an own goal from San Diego goalkeeper Pablo Sisniega three minutes later and a second White goal on the stroke of half-time made it 3-0.
San Diego raised hopes of an unlikely fightback when Mexico striker Hirving Lozano lashed in a long-range effort in the 60th minute.
But Vancouver continued to look the likelier team to score and Sisniega was shown a red card for bringing down Ryan Gauld when the Whitecaps player rounded him to go in on goal.