Del Potro took control when he broke Dolgopolov to go up 3-1 in the third set and then broke him again in the final game.
The Argentine will face sixth-seeded Nicolas Almagro of Spain in the semis on Saturday. Almagro beat defending champion Kei Nishikori 7-6 (2), 5-7, 6-3.
"I got into the first service game right away and that allowed me to get into the match," Raonic said. "And I just got better and better."
Raonic is coming off a win in the Thailand Open last Sunday and will be seeking his third title of 2013. He lost in the final of last year's Japan Open to Nishikori.
Dodig beat Jarkko Nieminen of Finland 6-2, 7-6 (3).