The Impact became the second team from the MLS - Real Salt Lake in 2010-11 - and first Canadian franchise to reach the CONCACAF Champions League decider after surviving an almighty scare at Estadio Alejandro Morera Soto.
Alajuelense netted four second-half goals but it was not enough as Montreal triumphed 4-4 courtesy of the away goals rule.
The Impact went into the second leg with a two-goal advantage following their 2-0 victory in Montreal on March 18.
And Frank Klopas' men added to the tally three minutes before the break in Alajuela, Costa Rica via Jack McInerney.
With a mountain to climb, Alajuelense pulled one goal back less than two minutes into the second half after Pablo Gabas fired a free-kick into the top corner of the net from the edge of the area.
Gabas was at it again on the hour-mark, tapping home from close range to give the home side a 2-1 lead.
Alajuelense's joy was short lived as Andres Romero restored parity on the night with a 72nd-minute strike past goalkeeper Dexter Lewis.
The Costa Ricans netted two unanswered goals during the final 11 minutes as substitute Allen Guevara and Jonathan McDonald set up a nervy finish but the Impact held on for a memorable aggregate victory.
Montreal will face the winner of America versus Herediano in the two-legged final.
