The 2002 champion conceded no break point chances to his 47th-ranked opponent and won in 59 minutes.
Nalbandian, ranked eighth, broke to lead 5-4 and took his first set point with a forehand winner. In the second set, he broke Montanes in the first and seventh games and finished the match with an ace.
Earlier, 42nd-ranked Simone Bolelli of Italy beat seventh-seeded Tomas Berdych of the Czech Republic 6-4, 7-5.
It was another good day for Germany, after qualifier Benjamin Becker's defeat of fifth-seeded Stanislas Wawrinka late Monday.
Andreas Beck, a lucky loser in qualifying, beat Nicolas Devilder of France 6-4, 6-4 and Philipp Kohlschreiber beat his compatriot Denis Gremelmayr 6-4, 7-6 (5).
Top-seeded Roger Federer was due on court in his hometown tournament later Tuesday in a first round match against 86th-ranked American Bobby Reynolds.
Nalbandian's Davis Cup teammate Juan Martin Del Potro, seeded third, was to face George Bastl of Switzerland.
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