The second seed is bidding for a trophy at his fifth straight Masters 1000 after ending last season as winner at Shanghai and Paris and sweeping Indian Wells and Miami last month.
He won the last 11 games in a row, finishing in 45 minutes and improving to 17-2 on the season.
The Serb ranked second in the world behind top seed and eight-time Monte Carlo champion Rafael Nadal ran his record over Spain's Montanes to a comfortable 6-0.
Ninth seed Jo-Wilfried Tsonga joined Djokovic in the third round with a hard-fought 6-4, 1-6, 6-4 victory over German Philipp Kohlschreiber on his third winning chance.
French compatriot Michael Llodra upset slumping Polish 16th seed Jerzy Janowicz 6-4, 6-2.
France's Julian Benneteau advanced over Austrian Jurgen Melzer 6-4, 3-6, 6-4 but Russian Teimuraz Gabashvili handed a 4-6, 6-4, 6-4 loss to Gilles Simon
Bulgarian 12th seed Grigor Dimitrov reached the second round over Marcelo Granollers 6-2, 4-6, 6-2 while Italian Andreas Seppi defeated 13th seed Mikhail Youzhny 6-3, 7-6 (7/4).
Tsonga needed just under two hours to go through in front of an enthusiastic crowd, with the seed taking a medical timeout on the way to finally securing victory on his third match point.
The winner managed four aces and broke Kohlschreiber twice from seven chances.
Tsonga now leads the series against the German number one 7-1, with his only loss occurring in 2009.
Tsonga is playing the tournament for a fifth time, with a 2013 semi-final his best result at an event the French consider a "home" tie. He continued his tradition of never losing in a Monte Carlo opening match.
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