Joao Neves’ early strike sent PSG ahead and then they scored three more in a flurry before half-time to take an unassailable lead.
Former PSG and Barcelona star Messi impressed in the second half as Miami battled well but could not find a way through.
“It is difficult to play against players of this calibre, it was difficult, but I think as a team we did an excellent job,” Luis Enrique told reporters.
“There’s a visible difference between the first and second halves… their quality is evident, and I think we deserved the victory.
“It was harder than it may seem, after the first half.”
The Spanish coach said his PSG team, who won the Champions League, Ligue 1 and the French Cup to claim their first ever treble, are in spectacular form and are hoping to triumph in the United States.
“Clearly, we are experiencing a historic season for our club, we aim to go far in this competition,” continued Luis Enrique.
“We are in the quarter-finals, with only eight teams remaining. Our goal is to progress as far as possible and to compete for the trophy. But for that, we need to overcome two more knock-out rounds.
“The season is very long, as it is for all teams, but the motivation to be competitive is there.”
PSG winger Ousmane Dembele said he was “glad to see” Messi and his former Barcelona team-mates Luis Suarez, Jordi Alba, Sergio Busquets and Javier Mascherano.
