The French fell victim to a very realistic Monteiro, since he converted the two break points he got, one in each set, without conceding any.
Among other notable results in the men's draw, Briton Ryan Peniston, 147th player in the world, overthrew Holger Rune (29th), quarter-finalist at Roland-Garros.
The 19-year-old Dane, seeded N.8, had won the first set 6-4, but the new darling of British tennis fought to snatch the second set 7-5 on the tie-break before flying over the third act (6-1).
Last week, for his first ATP Tour tournament, at 26, at Queen's, Peniston offered the Dutchman Casper Ruud, finalist on Parisian clay, in the first round.
Peniston, who was offered a wild card to play Wimbledon next week, was then knocked out in the quarter-finals by eventual finalist, Serbian Filip Krajinovic.