There was very little to separate the two players in the opening set until the 11th game, when Murray broke for a 6-5 lead. The US Open champion then held his next service game to open up a 1-0 advantage.
Murray then recorded a double break in the second set to go 4-1 in front against his French opponent.
Mathieu responded with a break and a hold of his own to bring the scores back to 4-3 in Murray's favour.
However, a hold and then a third break for Murray saw the Scot record a 6-3 success.
He will now play Jerzy Janowicz for a place in round four.