The World No.14 needed an one hour and 13 minutes to beat the unseeded Russian, ranked 99th in the world.
It was Bautista Agut's first title at Chennai, in his second final after losing to Serbian Janko Tipsarevic in 2013.
The second seed broke Medvedev's serve in the fourth game of the first set to surge to a 3-1 lead and soon wrapped up the first set in 33 minutes.
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The 20-year-old Medvedev fought hard in the second set and was level at 4-4. But Bautista Agut broke his serve in the ninth game and then claimed the contest.