"Ivo has really improved a lot over the past few years so it's always nice to get through a tough match like this."
Murray, the world number four, will next play the 15th seed Tommy Robredo of Spain, who beat 19th seed Marin Cilic 7-5 7-6 (7-2).
"It was very good, apart from a couple of games - when I lost my serve in the first set and had to save break points in the second," said the British number one.
"But I served well when I needed to and it was certainly better than my first match."
Murray showed a big improvement on his performance against Albert Montanes in the previous round, when he still looked to be suffering from the debilitating virus that kept him out of the Davis Cup defeat by Ukraine.
Mathieu troubled Murray with a series of drop shots but only converted one of the four break points he earned, failing with three in the seventh game of the second set, where Murray broke to lead 3-1 with a deft lob.
Murray stretched his lead to 5-2 and, on his first match point, a strong return of serve forced the Frenchman to net a backhand to hand him the match.
"I felt good, but physically I need to get better," said Murray.
"I thought I might be a bit slow, but my movement was good. I am not making many mistakes but I am not hitting the ball that hard.
"That will be better by the end of the week, if I am still here, or in Miami. But I am not putting myself under any pressure."
In other third-round action on Monday, there were surprise defeats for Frenchmen Jo-Wilfried Tsonga and Gilles Simon
Russia's Igor Andreev defeated 11th seed Tsonga 7-5 6-4, and Ivan Ljubicic beat eighth seed Simon 6-3 7-6 (7-3).
Fernando Gonzalez of Chile, the 17th seed, ousted American 13th seed James Blake 7-5 6-1, while Spain's Fernando Verdasco had an impressive 6-3 6-2 victory over Richard Gasquet of France.
In the women's event, fifth seed Ana Ivanovic eased to a 6-3 6-3 win over Gisela Dulko of Argentina, while seventh seed Agnieszka Radwanska crushed Canadian Aleksandra Wozniak 6-1 6-4.
"I had lots of doubts about my game in the past six months," said Ivanovic. "So finally I feel like I have strategy and also a plan in place again. It gives me a lot of security and confidence."
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