The world number 47 was leading 3-2 in the second set before Sakkari drew level at 4-4.
Sixth seed Sakkari reeled off the next two games to force a decider before claiming the victory with a forehand winner, guaranteeing a surge into the WTA's top 50 when the new rankings are released on Monday.
The 23-year-old fell on her back on the red clay and went on to kiss the surface in celebration after being congratulated at the net by Konta.
It was also a positive week for Konta's French Open preparations as her four victories this week are the first time she has claimed more than two WTA main draw wins in a row on clay in her career.
(Reporting by Hardik Vyas in Bengaluru, editing by Pritha Sarkar)
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