Drogba's goal in the fourth minute of injury time came after a terrible blunder by Galaxy goalkeeper Brian Rowe, who allowed the former Chelsea striker's long-range freekick to squirt from his grasp and into the net.
The injury-time strike was the final twist of a pulsating game in Montreal which left the Canadian club firmly among the Eastern Conference front-runners.
Giovani Dos Santos had fired Los Angeles into the lead after only eight minutes, capitalising after Maxim Tissot failed to head clear to leave the Mexican international a simple tap-in.
Montreal hit back though on 26 minutes with a superbly worked counter-attack which saw Dominic Oduro sprint almost the length of the field before passing to Ignacio Piatti who tucked away the equaliser.
The Impact took the lead early in the second half with a fine individual goal from Lucas Ontivero, who lost his marker with a clever turn before unleashing a ferocious shot into the top corner in the 56th minute.
Los Angeles responded within two minutes when Mike Magee controlled Steven Gerrard's cross into the area, turned and drilled in for 2-2.
That looked to be enough for a share of the points until Rowe's late gaffe gifted Montreal the win.
Elsewhere Saturday, the New York Red Bulls backed up last week's demolition job on New York City FC with a 3-0 win over Toronto FC which came courtesy of a quickfire hat-trick from Bradley Wright-Phillips.
The English striker grabbed his treble inside the first half hour, scoring in the 4th, 25th and 27th minutes to sink Toronto.
The result left the Red Bulls level on 19 points with Montreal, one behind Philadelphia Union, who took a one-point lead at the top after battling to a 1-1 draw against the Colorado Rapids in Denver.
Philadelphia looked to be heading to defeat after Sam Cronin struck for the Rapids in the 87th minute.
But Brian Carroll pounced in the second minute of injury time to grab a share of the points for Philadelphia.