Murray, Williams advance in Miami
Murray, who won this event four years ago, needed one hour and 54 minutes to oust the 29th-seeded Dimitrov.
Murray said he got off to a slow start but soon figured out how to cause problems for his opponent.
"I was leaving the ball a bit short. Once I started to improve my depth a bit I made it tough for him and he started to make mother mistakes after that," Murray said.
Dimitrov, last year's runner-up to Novak Djokovic, would have faced a tougher task had not the 21-year-old lost his composure when serving for the first set at 5-3.
Dimitrov double-faulted three times to allow Murray to break back and the Scot duly prevailed in the tie-break.
It is the second time in as many tournaments that Dimitrov has cracked when serving for a set against a top-ranked player, serving up four double faults when he led world number one Djokovic 5-3 in the third round at Indian Wells.
"He probably got a little bit nervous, and that helped me," Murray said.
Murray, who committed 22 unforced errors to Dimitrov's 38, will face Italian Andrea Seppi in the fourth round, the 16th seed having beaten experienced Brazilian Thomaz Bellucci 7-5, 4-6, 6-2.
Meanwhile, eighth-seeded Frenchman Richard Gasquet rolled over 28th-seeded Russian Mikhail Youzhny 6-3, 6-4 and 10th-seeded Spaniard Nicolas Almagro beat Belgian David Goffin 6-3, 6-4. Gasquet faces Almagro in the fourth round.
Also, sixth seed Jo-Wilfried Tsonga, of France, cruised past Finland's Jarkko Nieminen 6-3, 6-3 as Canada's Milos Raonic withdrew from the tournament with strep throat just moments before his match against Sam Querrey. In the women's draw, China's Li Na set up a quarter-final against world number one Serena Williams with an error-strewn 7-6 (8/6), 6-2 win over giantkilling Spanish teenager Garbine Muguruza.
The 31-year-old Li is playing in her first tournament since suffering an ankle injury in January's Australian Open final.
Williams also struggled having to come from a set down to win 2-6, 6-4, 6-2 against Slovakia's 13th seed Dominika Cibulkova.
Williams, who is bidding for a record sixth Miami title, admitted she had not been at her best and at 4-1 down to Cibulkova in the second set and staring defeat in the face she had had to pull herself together.
"I was struggling mentally, I was making so many errors," said Williams, who regained the number one spot in Doha in February but has not played since.
"I could not pull myself together. Someone (a fan) kept telling me to relax and I listened to him."
Li - the 2011 French Open champion and two-time Australian Open finalist - made 39 unforced errors, including four double faults, and allowed her 19-year-old opponent to battle back to 5-5 in the first set having led 5-1.
Even at 6-1 in the tiebreker Li faltered as 73rd-ranked Muguruza, who had claimed the scalps of the seeded Caroline Wozniacki and Anastasia Pavlyuchenkova in previous rounds, fought back to 6-6.
However, fifth-seeded Li pulled herself together to take the next two points and the set, before wrapping up victory in the second set.
Meanwhile, defending champ Agnieszka Radwanska defeated American Sloane Stephens 4-6, 6-2, 6-0 to reach the quarter-finals.