The British number one lost the opening set having held a 4-0 lead but recovered impressively, showing the significant improvements she has made on the surface to win 6-7 (3) 6-4 6-1.
Konta's only previous victory over a top-10 player on clay came against Roberta Vinci at the same venue three years ago.
— WTA (@WTA) May 16, 2019
Wednesday's wash-out means Konta must return to the court later on Thursday to take on Venus Williams, who was given a walkover when her sister Serena withdrew on Tuesday.
Konta has had by far her best season on clay, reaching the final in Rabat earlier this month and looking comfortable on the surface in a way she has not in the past.
Although she won all of the first four games, they were not easy for Konta and, as she began to make more errors, the tide turned, with Stephens winning six of the next seven games.
Konta broke back to force a tie-break but it was Stephens who was the more solid.
The second set was nip and tuck, with former US Open champion Stephens recovering an early break, but Konta forged ahead again to lead 5-4 and served it out to level the match.
Stephens had smashed her racket on the court in the opening set and thought about doing it again when Konta broke in the third game of the decider, and the American grew increasingly wayward as her opponent finished the match on a run of six games in a row.
Konta, whose only other top-10 win this season came against Stephens in Brisbane, will now try to reach the quarter-finals in the Italian capital for the first time having fallen in the third round for the last three years.