The match began quietly as neither side was able to create any clear-cut chances or threaten the opposition net for the first quarter of an hour.
However, the game suddenly sparked into life when Drogba latched onto Nigel Reo-Coker's cross before slotting the ball past Sean Johnson in the 27th minute.
Chicago were level just nine minutes later when Jeff Larentowicz converted from the penalty spot after Max Tissot brought down Harry Shipp in the area. Montreal then retook the lead in the 42nd minute after Wandrille Lefevre's header found the bottom corner of the net.
Chicago continued to threaten in the second half and they took the lead for the first time when Kennedy Igboananike headed in from a corner just before the hour mark. However, their lead lasted only two minutes as Drogba capitalised on some poor defending to score his second on the night.
The former Chelsea man completed his hat-trick after 65 minutes as the visiting defence failed to stop him from heading the ball into the net.
Both sides continued to create chances for the remainder of the match but Montreal held on to secure a big three points.