It is a first ATP quarter-final for Edmund, 20, who next plays world number six Tomas Berdych.
Yorkshireman Edmund, ranked 102nd in the world, is projected to better his career-high ranking of 99th when the new standings are released next week.
He is joined in the quarter-finals by world number one Novak Djokovic.
The Serb beat Fernando Verdasco 6-2 6-2, while world number five Rafael Nadal progressed with a 6-3 6-2 win over Robin Haase.
British number three Kyle Edmund beat Spain's Daniel Munoz de la Nava 5-7 6-3 7-6 (7-3) to reach the last eight of the Qatar Open.