The Spaniard, playing the event for the sixth straight year, defeated Croatian Borna Coric 7-6 (7/4), 6-4 in just under an hour and three-quarters.
The 23rd-ranked winner goes through to a Friday showdown with either Greek youngster Stefanos Tsitsipas or Tunisian giant-killer Malek Jaziri, who knocked world number four and top seed Grigor Dimitrov out in the first round.
French second seed Lucas Pouille was due to face the Japanese eighth seed Yuichi Sugita later.
Bautista Agut overcame six aces from Coric, who made his name in Dubai three years ago when he knocked out Andy Murray.
"I am happy, I played very good tennis," Bautista Agut said. "It was not easy to beat Borna. He's very solid.
"He was playing very good serves. He was playing very aggressive -- it's my best match of the tournament.
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