The defending champion looked impressive from the off but had a scare when he called for the trainer in the first set although he still closed out after breaking Tsonga early.
The second set was a straightforward affair as Murray eased to a 6-4 6-1 victory and set up a meeting in the next round with Novak Djokovic.
Murray felt the key to the win was taking his opponent out of his comfort zone. "I served well. I didn't give him any opportunities on my serve at all, and then as soon as the ball was short I attacked him," said the Scot. "I came forward, I kept him away from the net which is where he likes to be. That's when he plays his best tennis, when he's the one coming forward, so I kept him pushed back on the baseline. I made very few unforced errors and it was a good match."
Murray broke in the third game and was playing some impressive stuff as he moved into an early 3-1 lead before he appeared to run into trouble in the fifth. The Scot had five break points but lost the game and appeared to be in some discomfort.
The next two games went with serve before the Wimbledon champion called for assistance and, although he spoke with the trainer, he did not have any treatment and when the action resumed the remainder of the set also went with serve.
In a repeat of the first set, Tsonga was again broken in the third game and Murray showed no signs of his earlier problems as he largely dominated proceedings with his forehand. That domination continued as Murray won all the remaining games, at one point taking 14 points in a row, to cruise into the last eight.
The quarter-final against Djokovic will be the first time they have met since the Wimbledon final and Murray feels he is getting to his best form at the right time.
"I'm looking forward to it; it's a great test for me," he added. "I think my game is kind of close to where I want it to be right now and he's obviously playing very good tennis again, winning last week and he's been pretty comfortable so far here. It's always a fun match up against him and we've played here a few times so it'll be another tough one."
Murray also confirmed he will speak to his medical staff about the discomfort in the first set, although he does not believe it held him back.
"I'm not sure, I haven't had it assessed yet so I'll go and see my physio when I get off the court, have a chat about it, and see what the problem is. I was still moving well, it was just painful. I thought I moved pretty well at the end at the match and loosened up as it went on."
Djokovic, a three-time winner of the Miami title, beat the Spaniard Tommy Robredo 6-3, 7-5 to join Japan's Kei Nishikori and Alexandr Dolgopolov of Ukraine in the last eight. Nishikori toppled the fourth-seeded Spaniard David Ferrer 7-6 (9-7), 2-6, 7-6 (11-9) in a three-hour battle while the on-form Dolgopolov ousted the Australian Open champion and third seed, Stanislas Wawrinka of Switzerland 6-4, 3-6, 6-1.
Djokovic had only two break points the entire match and that was all he needed, converting both.
"It was a very solid win," said Djokovic. "I just was telling myself to stay mentally tough out there and composed and not get carried away by few points. I have done well. I have served well. I made him play an extra shot in important moments and that's why I'm satisfied with the overall match today."
It was the second match of the tournament for Djokovic, who received an opening-round bye on a walkover when Florian Mayer withdrew with a groin injury.