The Scot, who made the semi-finals here last year, broke serve in the first game and never looked back, wrapping up a straightforward 67-minute victory when Troicki sent a backhand wide.
"It was very good, getting off to a good start in the match," Murray told a news conference.
"Sometimes it's close, tight, first set, first match on clay, kind of anything can happen. You can start rushing. It's very easy on the clay to start rushing and making mistakes but because I got ahead I didn't need to do that."
Third seed Murray, who received a first-round bye, looked in fine form as he unsettled Troicki with some exquisite drop shots and played his usual tight defensive game to deny his opponent any chance to come back.
"I moved well. I was sliding pretty well on the court. Normally that's the thing that takes time to get used to... For me that's a good sign on the clay, whether I'm playing well or not," the Briton said.
Troicki took only five points on Murray's serve and never threatened the Scot who will take on Frenchman Julien Benneteau or Austrian 15th seed Juergen Melzer for a place in the quarter-finals.
Spaniard Fernando Verdasco, the 13th seed and 2010 finalist, battled past Croatia's Ivan Dodig 3-6 6-3 6-1 to set up a third-round clash with Frenchman Jo-Wilfried Tsonga, the fourth seed.
Tsonga pleased the crowd, who sang in celebration of his 27th birthday after he defeated German Philipp Kohlschreiber 6-2 6-4.
"Verdasco is far from being a present," Tsonga said.
"But on the other hand, it's a gift because I'm going to confront one of the best players on clay court.
"It's always good to play against the top players on clay because you can get used to the high quality of their game and you will not be surprised when you play them in major events."
Argentine Juan Monaco's fine run of form ended abruptly after he was forced to retire with a sprained ankle.
The 11th seed, who triumphed at the U.S. Claycourt Championships on Sunday, collapsed to the ground after he rolled his ankle while chasing down a Robin Haase shot at full stretch.
Monaco tried to continue after taking an injury time out to strap his ankle but quit while leading 5-7 6-0 3-2.
Tenth seed Feliciano Lopez succumbed to Swiss Stanislas Wawrinka 6-1 6-4.
Japan's Kei Nishikori, the 12th seed, took advantage of a rain interruption to recover from a break down in the second set and beat Spaniard Albert Ramos 6-2 7-5.