The Serb - who has endured two gruelling five-setters this week - was a set and two breaks down, but had been severely hampered by an injury to his left foot, which required heavy strapping.
Almagro was in control from an early stage, securing two breaks of serve on his way to wrapping up the set 6-2.
The Spaniard then broke early in the second to take full control of the contest.
Tipsarevic did force a break point in the second set, but could not profit from it. The world number nine called it a day after Almagro moved 5-1 in front.
The 10th seed will now face David Ferrer in the last eight.