Murray broke the Ukrainian on his second service game to move into an early 3-1 lead, before breaking again in the seventh after twice going to deuce.
Murray then took his final service game to love to win the opening set in just 28 minutes.
The second set started in the exact same way, with Murray breaking in the third game, but Dolgopolov fought right back for his first break to pull it back to 2-2.
However, Murray did not lose another game in the match, breaking twice more on his way to a 6-2, 6-2 win in under an hour.
Murray faces Radek Stepanek, who defeated John Isner, in the quarter-finals.