The fourth-ranked Murray easily won the first set but then began to miss more first serves and commit more unforced errors, while the 58th-ranked Monaco won several points with an effective drop shot.
Murray blamed changing conditions for the way the momentum swung, as the courts dried out after two days of rain.
"To be fair, he started playing a lot better,'' Murray said. "It wasn't my best match, but I was still very close to winning.''
Murray's other losses this year came against top-ranked Rafael Nadal - twice in Masters Series events in Indian Wells, California, and Monte Carlo - and to Fernando Verdasco at the Australian Open.
Monaco will next face 15th-seeded Marin Cilic, a 1.98-meter (6-foot-6) Croatian who defeated Christophe Rochus of Belgium 6-0, 5-7, 6-2.
Nadal and Roger Federer both advanced in straight sets. Nadal took the crowd out of play with a methodical 6-2, 6-3 win over Andreas Seppi, the top-ranked Italian at No. 37, while Federer beat big-serving Ivo Karlovic 6-4, 6-4.
In another upset, Jurgen Melzer of Austria eliminated seventh-seeded Nikolay Davydenko 7-5, 7-6 (5).
Davydenko missed two months earlier this season with a left heel injury, but he reached the quarterfinals in Monte Carlo and the semifinals in Barcelona the last two weeks.
In the first round, held over from Tuesday due to rain, Richard Gasquet beat his fellow Frenchman, ninth-seeded Jo-Wilfried Tsonga, 7-6 (2), 6-4.
In another first-round match, Paul-Henri Mathieu of France eliminated 11th-seeded David Ferrer 6-3, 2-6, 6-2. Ferrer was the runner-up to Nadal in the Barcelona Open final Sunday.
Gasquet then beat Ernests Gulbis of Latvia, 6-2, 1-6, 6-4, in his second match of the day, while Mathieu lost to Mischa Zverev of Germany, 3-6, 7-6 (5), 6-0.
Also advancing was fifth-seeded Juan Martin Del Potro, who dispatched Viktor Troicki of Serbia, 6-3, 1-6, 6-3.
The eighth-seeded Verdasco eliminated the last American in the draw, Mardy Fish, 6-4, 5-7, 6-3, and No. 12 Fernando Gonzalez beat Janko Tipsarevic 6-4, 6-4.
Seppi beat Nadal on a hard court in Rotterdam last year, and he had the fans cheering when he broke the Spaniard in the opening game, but the top-ranked player broke right back and negated Seppi's attacking game.
Nadal extended his winning streak on clay to 26 matches, stretching back to an opening-round loss to Juan Carlos Ferrero in Rome last year. Nadal won this clay-court warmup for the French Open three consecutive times from 2005-07 and he is trying to become the first player in the Open era to win it four times.
Thomas Muster also won three Rome titles, in 1990, '95 and '96.
The 2.08-meter (6-foot-10) Karlovic had 15 aces to Federer's one, but Federer still managed to break the big Croat early in each set.
"I got off to a good start in both sets, which is always good, kind of comforting against Ivo,'' Federer said. "He found his groove later on his serve, but I already had the break in the second, which is perfect.''
Federer saved all three break points he faced and committed only 16 unforced errors to Karlovic's 38.
In his last tournament, Federer was beaten by Davis Cup teammate Stanislas Wawrinka in the third round in Monte Carlo.
"I can't judge this kind of kind of a match because it's not real tennis with Ivo because it's all about reaction,'' Federer said. "Next match will be a different type of a clay-court test.''
Federer will next face Radek Stepanek, who eliminated the 13-time Grand Slam winner in last year's quarterfinals. Stepanek also won two matches Wednesday, playing five sets in all.
