Pinnington Jones claimed his first match win at a Grand Slam with victory over Tomas Martin Etcheverry at Wimbledon last year, before losing to Cobolli in the second round.
After overcoming the Italian in Dallas, the 22-year-old will face Australia’s James Duckworth or American wildcard Eliot Spizzirri next.
A close friend of British number one Jack Draper, Pinnington Jones completed his junior season at Texas Christian University in May but opted against returning for a final year in order to pursue a full-time tennis career.
Elsewhere, Novak Djokovic has withdrawn from the upcoming Qatar Open in Doha because of fatigue, according to tournament organisers.
The 38-year-old Serb will next target the North American hard court swing after finishing runner-up to Carlos Alcaraz at last month’s Australian Open.
Elena Rybakina won her first match since her Australian Open triumph with a 6-2 6-4 victory over China’s Wang Xinyu in Qatar, while world number two Iga Swiatek breezed past Janice Tjen 6-0 6-3.
But American world number five Coco Gauff lost 6-4 6-2 to Italy’s Elisabetta Cocciaretto, who progressed to the last 16 of a WTA 1000 event for the first time.
Britain’s Emma Raducanu retired from her first-round match against Camila Osorio earlier in the week.
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