Miami coach Gerardo “Tata” Martino opted to rest several of his first choice starting line-up, with Messi omitted from the squad altogether and Luis Suarez and Sergio Busquets on the bench.
But the wisdom of that decision soon came under scrutiny after Montreal dominated the early exchanges before Alvarez headed the visitors into the lead from a corner in the 13th minute.
Miami slowly worked themselves into the game thereafter, and Jordi Alba had the ball in the net in the 17th minute only for it to be disallowed for offside.
The hosts came close to an equaliser just before half-time after Julian Gressel’s superb through ball released Robert Taylor.
Finland international Taylor deftly lifted his finish over advancing Montreal goalkeeper Jonathan Sirois, only to watch as it crashed back off the cross bar.
Taylor should have equalised moments later but his header from Tomas Aviles’ inviting cross went straight at Sirois who gathered easily.
Miami continued to carve out openings in the second half, and had the ball in the net in the 59th minute when Taylor was adjudged offside after setting up Leonardo Campana.
But Miami finally got back on level terms in the 71st minute, with Campana heading home at the back post after a pinpoint cross from Lawson Sunderland.
Yet Miami’s hopes of preserving their unbeaten start to the season were left in tatters after Montreal scored twice in three minutes soon afterwards.
Coccaro made it 2-1 after heading home Mathias Choiniere’s cross, and then Nigerian forward Sunusi Ibrahim made it 3-1 with a deflected effort after a sweeping Montreal counter-attack.
Alba raised hopes of an Inter Miami fightback with a fabulous 80th minute strike from the edge of the area that curled into the top corner, but despite a late Miami onslaught, the visitors held on to take all three points.
In other MLS games on Sunday, Cincinnati were held to a 0-0 draw at home to D.C. United, while in Nashville, Los Angeles Galaxy recovered from 2-0 down to snatch a 2-2 draw on the road.
Teal Bunbury’s penalty and Dru Yearwood’s 58th-minute goal put Nashville 2-0 up, but Galaxy fought back to draw with goals from Riqui Puig and Dejan Joveljic.
© Agence France-Presse
Photo Twitter @davehybeee