British number one Konta appeared on course for her first tournament win in two years after cruising through the opening set and opening up a 4-2 lead in the second.
But sixth seed Sakkari, 23, hit back to twice break Konta and seal the second set 6-4 before romping to her first career title by taking the decider 6-1 and denying the Briton her first tournament success on clay.
Sakkari appeared beaten when she lost her serve in the fifth game of the second set, but after trailing 4-2 she rallied impressively to win 10 of the next 11 games.
Konta, 27, visibly tired as the Greek took control having spent three more hours on court than her opponent this week.
World number 47 Konta reached the first clay-court final of her career with a straight-sets victory over Australian fourth seed Ajla Tomljanovic in the last four.