The Czech Republic side had a clutch of players missing through injury and Covid-19 issues, including number one goalkeeper Milan Heca and teenage star Adam Hlozek, but they took the lead in the 26th minute through striker Julis, and he scored a second just before the interval.
Celtic substitute Leigh Griffiths pulled a goal back in the 64th minute, shortly after coming off the bench, and the Scottish champions, who had lost to AC Milan at home and drawn at Lille in their first two Group H fixtures, then piled on the pressure.
However, they were undone yet again in the 76th minute when Julis tapped in a breakaway goal before substitute Ladislav Krejci headed in a fourth in the final minute to complete a chastening night for Neil Lennon's side.
Celtic went into the game on a high after their impressive 2-0 William Hill Scottish Cup semi-final win over Aberdeen at Hampden on Sunday ended a rare four-game winless run.
Lennon kept an unchanged side, unsurprisingly, but it was a somewhat calamitous night for the Scottish champions, who came unstuck in a tumultuous spell midway through the first half.
David Moberg Karlsson, who had a short spell at Kilmarnock, had a drive saved by the foot of goalkeeper Scott Bain before he cracked the top of a post with another effort.
Bain did well to tip a shot from Julis over the crossbar for a corner, but from Moberg Karlsson's delivery, defender David Hancko sent a header against a post and from the rebound Julis, again unmarked, lashed the ball high into the net.
The goal shook the home side but they came close to levelling in the 38th minute when Mohamed Elyounoussi ghosted into the box to get on the end of a terrific cross by Ryan Christie but his cushioned volley clipped the top of the bar and went over, with stand-in keeper Florin Nita beaten.
Sparta's second goal, which arrived seconds before the break, came when Adam Karabec set up Julis inside the box and he neatly sidestepped Hoops midfielder Callum McGregor before curling the ball low past Bain and into the corner of the net.
The half-time interval gave Lennon the chance to sort things out, but Julis soon had the chance for his hat-trick.
Bain's slack pass was intercepted by Karabec who shifted it quickly to the Czech striker but he lofted his shot high over.
In the 63rd minute Sparta midfielder Michal Sacek drove into the Celtic box and smashed a shot off a post with Bain beaten and seconds later, Celtic reduced the deficit.
Griffiths played the ball to Tom Rogic and when his shot from inside the area was parried by Nita, the Hoops substitute raced in to slam the loose ball into the net.
But the visitors scored a classic counter-attacking goal as Moberg Karlsson skipped past Shane Duffy with ease to set up Julis a couple of yards out and the final nail came at the end when Borek Dockal curled a cross in for fellow sub Krejci to head into the net.