This is his sixth career top 50 win and second at a Grand Slam after knocking out France’s Ugo Humbert at Roland Garros this year.
Fearnley has now won on his main draw debut at each of the four Grand Slams – including runs to the third round at the
Australian Open and
Roland Garros this year.
This is still the 24-year-old’s first full calendar season on the ATP Tour, with the British star reaching a career-high ranking of world No.49 back in June.
"Even though I've had some pretty big match experience, it's still all relatively new for me, and being in these kinds of moments and these matches, it still gives me a lot of nerves.
“Just sometimes in those big moments, I need to be a little bit more clear and that will come with experience. It's not something I'm worried about.
"I'll be playing Zverev next on another big court - a high-stakes, high-tense environment and I think putting myself in those situations will help."
It was a promising start for the British star, who saved a set point to go on and break the 37-year-old Spaniard at 5-5 to take the opening set.
Fearnley seemed to be on the ascension – claiming a double break in a dominant second set and losing just five points on his serve to move just one set away from victory.
The match seemed all but wrapped up when Fearnley got the break in the third set and had the chance to serve out the match at 5-4, but more double faults helped Bautista Agut to a run of four straight games to force a fourth set.
While it would have been easy for the match to get away from him at that point, Fearnley responded well in the fourth. The Briton came back from 2-0 down to clinch back-to-back breaks and, with the chance to serve out the match again and 5-4, did so in emphatic style.
Fearnley moves on to play Zverev in the next round, in a repeat of the Australian Open third round earlier this year.
The two have faced each other twice this season, with the German winning in straight sets on both occasions.