The Scot was tested for much of the contest, but pounced on lapses of concentration from his young opponent at critical junctures in the match to advance in 89 minutes.
Murray had never faced the world number 76 on the ATP Tour before, and the 20-year-old served notice of his confidence with a no-nonsense opening service game featuring two aces.
Murray kept pace, but it was the Bulgarian who had the first chance to break, with the Scot needing to find some impressive passing shots to save three break points in the fourth game.
Dimitrov looked every inch a match for Murray in the opening games, but a sloppy game at 4-4, combined with some brilliance from the world number four, gave Murray a 0-40 advantage. Dimitrov could only deny him once before the Scot nudged ahead, and Murray's experience told as he closed out the set with a hold to love.
At 15-30 down in the first game of the second set, the Bulgarian dug deep to save back-to-back breaks.
Murray looked assured on his serve, moving well and offering very little chance for Dimitrov to attack him.
Dimitrov settled back down on serve too, and once again the players shared the first eight games.
But a double-fault from the Bulgarian to start the ninth was inauspicious, and having conceded a break point with a backhand into the net, he followed up with an errant forehand to hand Murray the crucial break.
The 20-year-old, perhaps sensing that his chance was gone, smashed his racquet to the ground in frustration.
He claimed a couple of points as Murray served for the match, but, in the story of the contest, the Scot won the points which mattered, wrapping things up when the Bulgarian's forehand went wide.