The fourth-seeded Djokovic suffered a second consecutive opening defeat, he also exited in his opener against Alejandro Tabilo in Monte Carlo two weeks ago.
Arnaldi, ranked 44th in the world, recorded just the second top-five win of his career and scribbled 'OMG' on the camera lens after his landmark victory.
"At the start I was just trying to not s*** in my pants, to be honest," said the 24-year-old Arnaldi.
The athletic Arnaldi squandered a 2-0 lead early in the contest but pounced again in the eighth game, breaking the Djokovic serve. The Italian wrapped up the opening set in 48 minutes.
Djokovic, a three-time champion in Madrid, fired 20 unforced errors in the first frame and finished the contest with a total of 32.
"He's my idol, he's always been. I was just glad that I could play him because I never played him, I just practised once with him. So playing him on a stage like this was already a victory for me," said Arnaldi, who showcased an entertaining brand of all-court tennis.
"But, you know, he is not at his best right now, so I came on court to try to play my best tennis and win. It happened. So right now, I don't even know what to say."
Elsewhere, Lorenzo Musetti, playing his first match since reaching the biggest final of his career in Monte Carlo, moved past Argentina's Tomas Etcheverry 7-6(3), 6-2 to book a third-round meeting with Stefanos Tsitsipas.
The talented Italian is on the cusp of making his top-10 debut, having entered the tournament at a career-high number 11.
Musetti admitted he is doing his best not to check the ATP live rankings website until he officially joins the top-10 club.
"This week, I don't want to look," said the 23-year-old, who moved up to number nine in the live rankings on Saturday.
Reigning Australian Open champion Madison Keys was broken while serving for her third-round victory at 5-4 in the second set against Anna Kalinskaya, but recovered to complete a 7-5, 7-6(3) in one hour and 48 minutes.
The fifth-seeded Keys lost to Kalinskaya earlier this month in Charleston.
"I knew I had to raise my level after getting a bit of a spanking last time," said Keys.
A semi-finalist in Madrid 12 months ago, Keys awaits Croatia's Donna Vekic in the round of 16, who defeated Emma Navarro 4-6, 6-3, 6-2 later in the day.
American fourth seed Coco Gauff eased past compatriot Ann Li 6-2, 6-3 to set up a last-16 clash with Swiss Olympic gold medallist Belinda Bencic.
Seventh-seeded teenager Mirra Andreeva continued her impressive form at WTA 1000 tournaments this season by storming into the fourth round with a 7-5, 6-3 performance against Polish 27th-seed Magdalena Frech.
The 17-year-old Russian, who won back-to-back titles in Dubai and Indian Wells, will take on Ukrainian qualifier Yulia Starodubtseva for a shot at reaching a second consecutive Madrid quarter-final.