Tsonga, who was serenaded with a brief rendition of "Happy Birthday" from the crowd, served five aces and broke his opponent three times.
"I had some worrying moments because I was almost broken several times," said Tsonga, who regularly used the drop shot to get out of trouble. "But I played well on the important points."
"Normally when I come here I usually find I need a few matches to get adjusted," said Murray, who lost to seven-time defending champion Rafael Nadal in last year's semifinals. "But I played very well, it was a very good first match."
Murray had spent a few days practicing on clay in Florida before heading to the Mediterranean.
"The sooner I can get on this surface for me the better," Murray said. "It was good that I stayed in the States and actually got sort of four or five days on the clay there before coming (here). I was sliding pretty well on the court. Normally that's the thing that takes time to get used to."
Murray didn't face a single break point in improving to 6-0 against Troicki.
Murray is being helped with his clay game by coach Ivan Lendl, who won three French Opens in eight Grand Slam titles plus two Monte Carlo Masters among his 94 career victories.
"He only arrived this morning so he's getting no credit for today," Murray said, joking. "But he'll help me the next five, six weeks."
Andujar had to rally from a set down to get past Federico Delbonis 3-6, 6-4, 6-2.
He was on the clay for several minutes, with Haase making a pillow for him with a towel. Monaco finished the fifth game but then had to quit.