For Djokovic, a three-time winner on the Miami hardcourts, the victory was a small measure of revenge as the rivals clashed for the first time since Murray beat the Serb in last year's Wimbledon final.
Murray, playing his first event since splitting with coach Ivan Lendl last week, had looked increasingly comfortable and confident on his own but Djokovic kept the Scotsman under almost constant pressure in blustery conditions.
"I was not surprised about the way he played," said Djokovic, whose coach Boris Becker was also not in Miami as he is taking time off to undergo double hip surgery. "I expected him to play well, to be a little bit more aggressive.
"I tried to not allow him to be in the comfort zone because when he strikes the zone, when he feels comfortable on the court, he's striking the ball so well, maybe best in the world."
The road to a fourth Miami title does not get any easier for Djokovic with a possible semi-final meeting with 17-times grand slam winner and two-times Miami champion Roger Federer up next.
Federer, who lost to Djokovic in the final of the BNP Paribas Open in Indian Wells 10 days ago, takes on Japan's Kei Nishikori in another quarter-final later on Wednesday.
In dramatic contrast to the women's semi between Dominika Cibulkova and Agnieszka Radwanska that ended moments earlier on centre court and featured 19 breaks of serve, breaks were hard to come by for Djokovic and Murray in a tight opening set.
Murray had only a single break opportunity against the Serb while Djokovic was able to convert one of his three chances, with the help of a controversial point.
The first set ended in an argument between Murray and the chair umpire, the Scotsman furious that Djokovic was given a point to open the decisive game even though it was clear the Serb reached across the net to hit the ball.
A distracted Murray then lost the next three points on unforced errors as Djokovic claimed the break and set.
"I wasn't sure, from where I was standing, it was a very hard thing to see," said Murray. "I knew it was close. So that's why I went and asked Novak and he told me he was over the net. That was it.
"(The chair umpire) said, 'yes, he was over the net but he was in line with the net,' so I didn't really understand.
"It maybe had a slight bearing on that game but I was still up a break in the second set."
Djokovic admitted later that he did indeed cross the net but claimed he was unaware it was against the rules because he did not touch the net.
"It's an important point but mentally obviously it distracted him more than it gave me an encouragement," said Djokovic. "It was him, that he got upset ... he made three unforced errors and the set was gone."