Edmund, who ended Andy Murray's near 12-year reign as British No. 1 in March, made the last eight at the Brisbane International and the Australian Open earlier this year, and he is now into another quarterfinal after overcoming an untidy start to beat Albot 6-,2 6-4.
Neither player could hold their serve in the first three games, but Edmund converted three break points to take the opening set in under 39 minutes.
And he sealed the match in the second set with a deep volley that forced an unforced error from Albot.
Edmund is now the highest seed left in the tournament, with No. 1 seed Albert Ramos-Vinolas exiting the tournament Wednesday.