The second seed was under the cosh for much of the opening set but turned things around to win 7-6 (7/3) 6-3 in one hour and 54 minutes.
Murray, who won the title here in 2009, struggled with his first serve from the off and found himself a break down in game four after some confident returning from his opponent.
The Scot broke back the very next game when Dimitrov missed an attempted lob but the Bulgarian opened up another lead straightaway, with a forehand volley at full stretch which left Murray stranded.
Dimitrov then saw two set points saved game eight as Murray clung on by his fingertips.
However, luck was on the Briton's side in the next game when Dimitrov, serving for the set, dished up three double faults to gift Murray a route back in.
The opening set was eventually settled by a tie break, where a couple of unforced errors from Dimitrov played right into the hands of the world number two.
That knocked some of the confidence out of the Bulgarian and he dropped game two of the second set, putting a weak forehand into the net.
Murray wasted two further break points in the sixth game but soon found himself serving for the contest.
And although his first two match points went begging, it was third time lucky for Murray as Dimitrov put a backhand into the net.
"It was obviously way cooler [with] much slower conditions," Murray told www.atptour.com. "The ball wasn't going as quick. 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 more mistakes after that."
Murray's next opponent will be Italian 16th seed Andreas Seppi, who also came through a tough encounter against Thomaz Bellucci 7-5 4-6 6-2.
Eighth seed Richard Gasquet eased into the last 16 with a 6-3 6-4 victory over Russia's Mikhail Youzhny while Spaniard Nicolas Almagro, the 10th seed, beat David Goffin of Belgium by the same scoreline.
Fourth seed Tomas Berdych saved two match points to beat Colombia's Alejandro Falla 2-6 7-6 (8/6) 6-4, but there was no such struggle for the sixth seed Jo-Wilfried Tsonga who won in straight sets.
Tsonga beat Finland's Jarkko Nieminen 6-3 6-3 while ninth seed Marin Cilic held off John Isner 6-3 7-6 (7/3).
Sam Querrey advanced as Milos Raonic withdrew due to illness.