The defeat meant that Rafa Nadal will return to the top of the world rankings on Monday.
World number 34 Coric, playing in his first grasscourt final, showed great mental toughness and determination to fend off two set points before clinching the opening set after 57 minutes.
Federer responded by applying further pressure on Coric's serve in the second set before converting a break point in the eighth game to force a decider.
The 21-year-old then broke Federer in the sixth game of the final set when the Swiss sliced a backhand long, before claiming only his second ATP title.
It was Federer's first defeat on grass since a second-round loss to German Tommy Haas at Stuttgart in June last year.
He also remains two matches away from matching Jimmy Connors' all-time record of 174 victories on the surface as he missed out on a 99th career title.
"Many congratulations Borna for your first grass title," Federer said in the post-match presentation. "You played a wonderful match and deserved to win at the end."
Coric is projected to rise to a new career-high 21st in the world rankings next week after collecting wins over the tournament's top two seeds, Federer and Alexander Zverev.
Federer, who lifted the Stuttgart title last week, will be bidding for a record-extending ninth men's singles crown at Wimbledon, which begins on July 2.
(Reporting by Hardik Vyas in Bengaluru; Editing by Ian Chadband)
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