The match was subject to three delays as the weather continued to play havoc with the schedule on Monday (Tuesday AEST), but the fifth-seeded Australian overcame the disruptions to win 7-5 7-6 (7-3) in two hours 22 minutes.
Fery, who shot into the world's top 50 after reaching the Wimbledon semi-finals at the All England Club as a wildcard, was playing his first match since that improbable run was ended by Alexander Zverev.
That catapulted the 24-year-old up the rankings to No.36 in the world, earning him a main draw second-round entry for the first time at an ATP 1000 Masters.
It nearly ended there at the Ohio tournament as another Australian, James Duckworth, put up a real fight before finally succumbing 6-4 2-6 7-6 (7-4).
De Minaur represented an even sterner challenge, though he was out of sorts early on, perhaps unsettled by an early rain break in his first service game, and had to save three break points in his next service game.
The match steadied until the 11th game when the world No.8 pounced, forcing a break point which he won with a firm drive that Fery returned into the net.
De Minaur then held to love to take the set, at which point the rain returned.
Upon resumption, the Aussie broke twice for a 5-1 lead before crumbling trying to serve it out and the contest was tied at 5-5.
But the 'Demon' got it done in the tiebreaker to set up a last-16 showdown with 21st seed Arthur Fils of France, a 6-1 6-7 (7-9) 6-3 winner over ninth-seeded Czech Jiri Lehecka.
"I'm happy with the way I found a way. It got tough at the end, probably a little bit harder than it should have been," said de Minaur, who last reached the fourth round in Cincinnati in 2019.
"I played a very good match until the last rain delay and then I had to dig deep. I had to kind of reset and take it one point at a time.
"I take confidence out of these types of tough wins and hopefully every day I feel better and play better."
Compatriot Rinky Hijikata won't be joining de Minaur in round four after he was beaten 6-3 6-4 by 14th seed Jakub Mensik.
A break in each set cost the Sydneysider and means Mensik next faces Novak Djokovic's conqueror, Argentine Thiago Agustin Tirante.
Elsewhere, there were wins for Zverev, Spanish teenager Rafael Jodar and Flavio Cobolli.
With PA.
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