"It's the best victory of my career," said Munar, who is ranked 77th in the world.
Fognini had Munar on the ropes in the final set and got to within two points of victory, but the youngster rallied, breaking the 31-year-old to level at 5-5 and winning with a burst of 12 points in a row.
He will battle local hope Guido Pella for a place in the semi-finals.
(Reporting by Rohith Nair in Bengaluru; editing by Sudipto Ganguly)
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