Alcaraz battles into Italian Open quarter-finals
Third seed Alcaraz had never lost a set in his previous four meetings with Russia's Khachanov but was forced to fight on the Foro Italico centre court.
Alcaraz needs to get to the Rome semis not just for a chance at his third title of the season but also to secure a top-two seeding at the French Open which begins later this month.
The Spaniard took his 12th clay-court win of the year to set up a clash in the last eight with fifth seed Jack Draper, a winner over France's Corentin Moutet.
"It was tiring.