The Serb, a five-time Rome winner, had been trailing 6-4, 2-1 when the game was halted overnight
World number one Novak Djokovic survived a three-set battle with Greece's Stefanos Tsitsipas over two days in their rain-delayed quarter-finals to advance to the Italian Open semi-finals on Saturday.
The defending champion won through 4-6, 7-5, 7-5 after 3hr 16min on court in the clash which had been suspended from Friday because of rain in the Italian capital.
The Serb, a five-time Rome winner, had been trailing 6-4, 2-1 when the game was halted overnight.
Djokovic twice recovered from a break down in the final set to seal a berth in the final four for the eighth straight year on his second match point.
The 33-year-old next meets Italian Lorenzo Sonego who shocked Russian seventh seed Andrey Rublev 3-6, 6-4, 6-3 in 2hr 33min.
"It felt like we played two matches, actually we did, one yesterday where he was a better player," said Djokovic.
"He started also better today I managed somehow to hold my nerve, breaking serve at important moments.
"Without a doubt, until the last shot I didn't know whether I was going to win but I believed that I could.
"I am just really, really glad to overcome this challenge. It was probably the toughest match of the year for me so far."
Monte Carlo winner Tsitsipas had moved to a 4-2 lead in the second set on their return to court Saturday, but Djokovic made a crucial break for 4-4, the Serb saving break points in the following game for 5-4 and breaking in the 12th game for 7-5 to level at one set all.
Both hit 15 winners in a balanced second set but Tsitsipas paid for four more unforced errors than his rival who used his forehand to break through with winners along the line.
Tsitsipas twice led by a break in the deciding set and served for the match at 5-4, but winner Djokovic rallied to take a 6-5 lead.
Tsitsipas served to force a final-set tie-break, but the 18-time Grand Slam winner again proved too strong for the Greek who has won none of their three meetings on clay.
Djokovic, 33, stretched his winning record over Tsitsipas, 22, to 5-2.
The Serb's next opponent, 33rd-ranked Sonego, won their only previous meeting in the quarter-finals on hard court in Vienna last year.