The British number one saw off his 19-year-old opponent 6-3 6-7 (6) 6-1 in just under two and a half hours to set up a quarter-final against Chile’s Alejandro Tabilo.
Having claimed the opening set, Norrie had match point in the tie-break before Van Assche came through to bring the match level.
He was then broken by Van Assche in the opening game of set three before hitting back emphatically, rattling off six games in succession to secure the win.
Norrie is seeded 19th for the Australian Open ahead of the draw taking place in Melbourne on Thursday.
A video on social media showed Djokovic being hit by the object from the stands before falling to the floor.
The Serbian was helped out of the arena by security so he could receive medical treatment.
Djokovic later posted a message on X to assuage any fears, and thank fans for their support after the incident.
He said: "Thank you for the messages of concern. This was an accident and I am fine resting at the hotel with an ice pack. See you all on Sunday."
A statement from the Italian Open said: "Novak Djokovic has been accidentally hit by a bottle while signing autographs.
"This is not being treated as a deliberate act, but as an accident. A boy was calling out to get an autograph and the bottle fell out of his bag. Novak has been taken to the medical centre."
They later issued an update, which read: "Novak has undergone the necessary checks and has already left to return to his hotel. His condition is not a cause for concern."
The incident happened after Djokovic’s 6-3 6-1 win over Corentin Moutet when he approached supporters in the stands.
The world number one is due to play Chile’s Alejandro Tabilo in the next round on Sunday.
Tabilo stunned Djokovic with a straight-sets victory in the third round on Sunday, winning 6-2, 6-3 in his maiden ATP Tour clash with the Serbian.
The Chilean advances to the fourth round at a Masters 1000 event for just the second time after Indian Wells in 2023, with Djokovic unable to explain his struggles in Rome.
"I just wasn't able to find any kind of good feelings on the court, to be honest, striking the ball. I was completely off," he said.
A six-time champion in Rome, the 36-year-old won his opening-round match at the ATP Masters 1000 event against Corentin Moutet.
However, Djokovic was inadvertently hit on the head by a falling water bottle when leaving the court after that victory.
Whether that incident impacted his performance against Tabilo remains to be seen.
"I don't know, to be honest. I have to check that," he said. "Training was different. I was going for [a] kind of easy training yesterday.
"I didn't feel anything, but I also didn't feel the same. Today under high stress, it was quite bad – not in terms of pain, but in terms of this balance.
"Just no coordination. Completely different player from what it was two nights ago. Could be. I don't know. I have to do medical checkups and see what's going on."
Tabilo's next opponent will be Karen Khachanov, who saw off Francisco Cerundolo 6-2, 6-4, while Djokovic is made to wait for his 1100th match win.
The Serbian, who has not played since the semi-finals in Monte-Carlo, made a slow start, falling 1-3 behind in the first set, but recovered to take the first match.
Moutet could not regain his level, and Djokovic dropped only one set as he cruised through the second match to advance after one hour and 26 minutes.
Djokovic, a six-time champion in Rome, will take on Alejandro Tabilo in the third round after the Chilean beat Yannick Hanfmann 6-3, 7-6 (7-4).
Data Debrief: Still on top
Djokovic now has 23 wins in ATP-1000 events since turning 35. He eclipses John Isner (22) for the outright second most wins in such tournaments since the introduction of the format in 1990; only Federer, with 51, has more.
The Serbian will have the opportunity to clinch his 1100th match win in the third round as he chases his first trophy of the season.
Tabilo recorded the biggest win of his career with a dominant performance, winning 6-2, 6-3 in his maiden ATP Tour clash with the Serbian.
Djokovic, who was hit on the head by a water bottle following his second-round win over Corentin Moutet, struggled to get going as he saw his serve broken four times.
Tabilo clinched the opening set in style, making few errors, and Djokovic failed to mount a comeback in the second, with double faults costing him at key moments, including one on the final break serve.
The Chilean advances to the fourth round at a Masters 1000 event for just the second time after Indian Wells in 2023. His next opponent will be Karen Khachanov, who saw off Francisco Cerundolo 6-2, 6-4.
Data Debrief: Big winner
Tabilo is the second player from Chile to defeat the World No.1 in ATP-1000 events after Fernando Gonzalez, who defeated Lleyton Hewitt at the Hamburg Masters in 2003, since the introduction of the format in 1990.
He is also the first Chilean to defeat a World No. 1 since Fernando Gonzalez beat Roger Federer at the 2007 Nitto ATP Finals Round Robin.
Djokovic, meanwhile, is made to wait for his 1100th match win.
The world number one made a slow start but found his feet midway through the opening set, getting the decisive break in the seventh game to inch ahead.
Tabilo only won one of 12 points on Sinner's first serve throughout the first set and saw his own serve broken again early in the second as Sinner eased to a 6-4 6-3 win.
Due to rain cancelling Friday's play, Sinner is on double duty on Saturday, with the Italian set to return to the court to face Andrey Rublev in the last eight at 7pm local time.
Data Debrief: Sinner joins exclusive club
Sinner's victory means he has reached at least the last eight at each of his first 10 ATP events of the season.
He is the first player to achieve that feat since Novak Djokovic in 2015, and the youngest – at 22 years and 358 days old – since Ivan Lendl in 1982.
Zverev was punished for a slow start as Tabilo took the opener within 32 minutes, but the German fought back in a one-hour, 15-minute slog of a second set.
The 2017 Italian Open champion put his foot down from 3-3 in the second-set tie-break, with Tabilo potentially feeling the effects of a fine run that included a last-32 upset of Novak Djokovic.
Zverev upped the intensity in the final set and won a huge 94 per cent of points behind his first serve as his opponent wilted, the third seed reaching the Rome final for a third time.
Data Debrief: Zverev draws level with Becker
Zverev will now face either Nicolas Jarry or Tommy Paul in Sunday's final, which will be his 11th at ATP 1000 Masters level.
That tally pulls him level with Boris Becker for most such finals by a German player since the format's 1990 introduction.
Zverev – who triumphed on the clay in Rome back in 2017 – produced a dominant serving performance against another heavy hitter in Fritz, but only after an awkward fall early on.
The German sought medical attention after falling on his stomach in the third service game of the match, the slip leaving him bleeding from both hands.
However, he soon shook off that knock and did not allow Fritz a single break point in a deeply impressive display, setting up a semi-final meeting with Chile's Alejandro Tabilo – the conqueror of Novak Djokovic – for Friday.
Data Debrief: Zverev gathering momentum
Zverev, the lone former champion remaining in the men's draw, is surely the favourite to win a tournament characterised by unfortunate withdrawals and shock defeats for the biggest names.
Seven of Zverev's 21 tour-level titles have come on clay, and he is yet to drop a single set in Rome this year. His three-year wait for an ATP 1000 title could soon be over.
