Clement started the second set in determined fashion but Murray never looked in danger on his own serve and a break at 1-1 saw him edge ahead.
Instead of counteracting the increased pressure Murray was applying, Clement's error count began to soar and a superb backhand drop shot gave the Scot a 3-1 lead.
Clement saved two more break points in the fifth game but it would be a brief reprieve as Murray brought up match point on his opponent's serve at 3-5.
The Frenchman saved with an ace but a searing forehand return brought up a second and another forehand winner saw Murray complete his triumph.
Murray was delighted with his performance, telling BBC Sport: "I played really well, much better than in the first round. I felt like I was more adjusted to the conditions and the result would suggest that.
"I served very well and when I gave him opportunities I came up with a few big serves at important moments."
Murray is coping with an ankle injury and added: "It changes every day. Some days it hurts round the joint, on some at the front and on some at the back. I am taking it each day at a time and making sure I get the right physio treatment."
His quarter-final against Gasquet is the last match on Centre Court on Thursday and he added: "We have had some very good matches in the past and I am hoping this will be another good one.
"The Wimbledon match last year was one of the most exciting of my career."
Top seed Novak Djokovic had to fight hard to beat Jan Hernych of Czech Republic 6-3 3-6 6-4 while third seed Gilles Simon beat Teimuraz Gabashvili 6-1 6-2.
David Ferrer, seeded four, earned a 6-4 6-2 win against Philipp Kohlschreiber and fifth seed Marin Cilic got the better of Julien Benneteau, winning 6-3 4-6 7-5.
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