Djokovic was on the ropes in the opening set against the 21-year-old Bulgarian, who raced to a 4-1 lead and served for the set at 5-3.
But Dimitrov served up four double faults in the ninth game and was broken, and that was the only opening that Djokovic needed.
“It was not really a beautiful match to watch,” Djokovic said. “There were a lot of unforced errors, and we both looked quite sluggish on the court, very slow,” he added.
He’ll now meet Sam Querrey, who edged Australian Marinko Matosevic 7-6 (7/5), 6-7 (7/9), 7-5.
Meanwhile, World No 3 Andy Murray, playing his first tournament since losing to Djokovic in the Australian Open final in January, got past tenacious Taiwanese Lu Yen-Hsun 6-3, 6-2.
The Scot next faces Argentina’s Carlos Berlocq, a 6-2, 6-2 winner over Japan’s Kei Nishikori. Germany’s Tommy Haas saved a match point en route to a 6-3, 6-7 (2/7), 7-6 (7/2) victory over Spain’s Nicolas Almagro.
Top seed and defending champion Victoria Azarenka and second-seeded Maria Sharapova led the way into the women’s quarter-finals of this combined ATP Masters and WTA event. Azarenka shook off a sore ankle to beat Urszula Radwanska 6-3, 6-1 and Sharapova stepped up her game in the second set to beat 87th-ranked Spaniard Lara Arruabarrena-Vecino 7-5, 6-0.