It was Don Matthews' fifth Grey Cup title as a head coach, tying him with Lew Hayman, Frank Clair and Hugh Campbell for the all-time record.
Leading 1-0 after 15 minutes, the Alouettes gained 99 yards on one play for what was, at the time, the longest touchdown reception in Grey Cup competition.
Terry Baker kicked a 42-yard field goal with 1:15 remaining in the second quarter to give the Alouettes an 11-0 lead at halftime.
With less than six minutes remaining in regulation and the Eskimos offence facing a third-and-10 on the Montreal 36, head coach Tom Higgins opted to go for it rather than attempt a 43-yard field goal.
Needing a two-point convert to tie the score, Tim Strickland broke up a pass intended for Terry Vaughn.
Edmonton tried one last-ditch effort with an onside kick attempt, but Copeland grabbed the football and ran 47 yards for his second touchdown, sealing the win for Montreal.