The second-seeded Murray fought back from a break down twice in the first set and then grafted his way through the second to take a commanding lead.
Murray failed to convert six match points against the towering Croat, who broke one back with the Olympic gold medallist serving for the match at 5-2.
But after sealing match point seven, Murray completed the straight-sets win against an opponent who made too many unforced errors to cause the highest remaining seed any serious problems.
Murray said: "Every time I was broken, I broke back almost straight away so I responded well despite it being a tough match against a good opponent in hot conditions.
"It was a tough, tough match. Second set especially was a lot of long, long games and tough points.
"I just managed to come through that. Obviously I took a while to get to the finish line but I thought I played a solid match with not too many errors and was tough in the important moments," he said.
The US Open champion is the highest seeded player left in the men's draw after world number one Novak Djokovic lost to Tommy Haas and will leapfrog Roger Federer to second place in the world rankings if he wins the title.
"I'd like to get to number two this week. It would be nice, but I'd rather focus on trying to win my matches," said the Scotsman.
Eighth seed Gasquet outclassed Tomas Berdych in the day's later quarter-final, the Frenchman recording a 6-3 6-3 victory to reach his first semi in Miami.
Gasquet was solid, hitting 17 winners and breaking serve three times as he made up for his defeat to Berdych in the round of 16 at Indian Wells.
The Frenchman knows he is the underdog against Murray, who he has lost to four times in seven matches, but believes he is capable of producing an upset.
"I can win against Andy. He's the favourite, the number three in the world but I know his game, I know how he plays," he said.
The other semi-final features David Ferrer against Haas.