Nik Antropov, Nikolai Zherdev and Ryan Callahan scored in regulation for the Rangers, who have won three of their last four games.
Henrik Lundqvist stopped 26 shots for the win.
Alex Kovalev, Maxim Lapierre and Markov tallied once each for the Canadiens, who have dropped three in a row.
Price took the loss despite a strong 39-save performance.
In a 4-on-4, the Rangers took a 1-0 lead 7:13 into the second period. Drury's initial drive from the right wing was kicked out by Price, but right to Antropov for a tap-in.
The Habs knotted the game only 1:34 before intermission, as Kovalev let fly with one of his trademark wristers for a power-play marker.
Zherdev opened the third-period scoring at 5:32 with a shot that changed direction off a Habs' player's stick, but Lapierre evened the score, 2-2, only 40 seconds later.
Callahan gave the Blueshirts a 3-2 edge with a little more than nine minutes left in regulation, but Markov forged another tie off a Guillaume Latendresse dish with 4:53 remaining in the third.
The Canadiens failed to cash in on a power play which bled into overtime, as Scott Gomez was whistled for tripping with 1:44 left in the contest.