Scheduled as the last match on the Grand Stand Arena, de Minaur spent a long afternoon and evening waiting to play his last-16 tie against the American due to unseasonal heavy rain, topped by a hailstorm.
Australia's No.1 started strongly, breaking in the opening game, but went down 7-5 6-3 in one hour 56 minutes for his first loss to Paul in six matches.
De Minaur, the seventh seed, was broken in the fourth and 12th games of the first set and was soon trailing 5-1 in the second.
Rome is the last big warm-up tournament before the French Open starts at Roland Garros on May 25.
11th seed Paul will next face Hubert Hurkacz of Poland, who beat Czech Jakub Mensik, seeded ten places higher at 20, 7-6 (7-5) 4-6 7-6 (7-5).
Earlier on Tuesday, Italian tennis fans were able to enjoy their returning hero Jannik Sinner secure progress to the quarter-finals.
The world No.1's arrival on Campo Centrale to play Francisco Cerundolo was also delayed and Sinner initially made heavy weather of the tie, the Argentine being in good form on clay this season.
But having taken the first set tiebreak, Sinner eased to a 7-6 (7-2) 6-3 victory in what was his third match back since a three-month ban for a doping offence.
"Today I felt that I raised my level," he said.
"It was a very long day … But I need this. I'm happy to be back. I was three months out, so every situation for me, I feel very fortunate to be here, very lucky."
Defending champion Alexander Zverev beat Frenchman Arthur Fils 7-6 (7-3) 6-1 and the German now plays Lorenzo Musetti, who had match point against Daniil Medvedev just as the rain arrived.
Nearly three hours later the pair came back on court and Italy's Musetti closed out with an inside-out forehand winner on the first point of the resumption.
Musetti, a new entrant in the top 10 of the rankings at No.9, won 7-5, 6-4.
Before the rains came Carlos Alcaraz moved into the last eight, and a rematch with Jack Draper who defeated him in the semi-finals at Indian Wells in March.
Alcaraz was made to work for a 6-3 3-6 7-5 victory over Karen Khachanov, his fifth in five matches against the Russian.
"Physically I was struggling a little bit," Alcaraz said.
"Not any pain in any part of the body. I was just tired. I had to run a lot. I was really proud the way I fought (for) every ball."
The third-ranked Alcaraz wore a long black brace that covered the upper portion of his right leg and stretched down to just below his knee.
Alcaraz withdrew from the Madrid Open last month due to an upper right leg ailment that bothered him during the Barcelona Open final. He also had a left leg injury.
Alcaraz, who won the Monte Carlo Masters in April, is now 12-1 on clay this season. The four-time grand slam champion will next face No.5 Draper, who rallied past Corentin Moutet 1-6 6-4 6-3 on the red clay courts of the Foro Italico.
With AP.
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