Six-time champion Djokovic was carrying an unspecified injury into the match as he lost for a second time to the rising 20-year-old Dane.
Rune defeated the Serb to win the Masters 1000 at Paris Bercy last November.
"This is probably the coldest, wettest tournament I've ever played here in Rome," the 22-time Grand Slam winner said.
"I don't recall so many days in a row (of) raining. In these conditions, it's very difficult to get the ball past him.
"He's a very talented, dynamic player - he was too good for me for most of the match," he said of Rune.
The latest contest between the pair was marred by spats with the chair umpire by both players - Djokovic on a time warning and Rune over a second-set line call.
Play was halted for just over an hour by rain.
Shortly after the match began, Djokovic requested extra towels to pad his lower back on the bench during most changeovers.
He has also suffered with a right elbow problem over the past weeks which caused him to miss the Madrid event and put his pre-Roland Garros preparation behind schedule.
He was treated by the trainer and tournament doctor after the third game of the second set and given a painkiller before playing on.
Djokovic was playing his 17th consecutive Rome quarter-final, now standing 13-4. Tomas Berdych was the last payer to beat him in the last eight in Rome back in 2013.
His Rome record dropped to 67-11 as his bid for a seventh title ended after two hours, 18 minutes with 35 unforced errors in only his fourth loss of the season.
"This is really a big win for me," Rune said. "Every match against Novak is a huge challenge. He's one of the greatest to ever play the game.
"I was proud of myself and enjoyed every minute out there," the seventh seed added.
"I have to stay humble, I still have a lot to achieve. I'm a huge fighter on court, I leave everything out there. I had to fight hard and play my best tennis.
"I'm a big fan of Novak; he's a huge inspiration. He puts huge pressure on you - I had to stay brave."
Rune will face either world number four Casper Ruud or the Argentine Francisco Cerundolo in the semi-finals.
