In a battle of two big servers, the American claimed the only break of each set to record his first hard-court victory over Almagro after 72 minutes of action.
Games were exchanged throughout the opener, with Almagro appearing the more likely to break, having missed two opportunities in the seventh game and held to love on his own serve three times.
However, it was Isner who went ahead by clinching the first set with a break in the 12th game.
The second set was a similar story, but this time it took Isner three attempts to get his break in the sixth game, with Almagro fighting in vain to try to avoid falling further behind in the contest.
There was little time for the Spaniard to battle back, though, as Isner fired down two comfortable service games on his way to securing a victory which sets up a last-eight meeting with Tomas Berdych.