Pressure was already on Neil Lennon following a run of one win in five games and his team were again below par against a Sparta side who were without a number of regulars due to coronavirus.
Julis opened his account in the 26th minute and doubled his tally on the brink of half-time with a curled finish, the goals coming either side of Mohamed Elyounoussi hitting the bar from a rare Celtic chance.
The host was given a route back into the game 25 minutes from time when substitute Leigh Griffiths made an instant impact, but Julis killed off the contest with his third and Ladislav Krejci added a late fourth to pile on the misery.
David Moberg Karlsson crashed an early shot against the woodwork and was denied with another effort by Scott Bain, who also kept out Julis with a brilliant reaction save.
However, Sparta's pressure told moments later as Julis lashed into the roof of the net after David Hancko's glanced header had come back off the frame of the goal.
Celtic orchestrated a chance 38 minutes in, Elyounoussi meeting Ryan Christie's pass and grazing the crossbar with his volley, but Sparta were soon two goals in front through Julis' fine finish into the bottom-right corner.
Michal Sacek was given space to fire a long-range shot against the post as Celtic came within inches of shipping a third, though that let-off acted as a turning point of sorts.
Florian Nita kept out Tom Rogic's shot but Griffiths was on hand to fire the rebound into an empty net against a Sparta side previously without a point to their name.
Just as Celtic had some momentum on their side, Karlsson got away from Shane Duffy and squared the ball for Julis to bundle in, condemning the hosts to a third successive home defeat in the competition for a first time.
Substitute Krejci then got on the end of a Borek Dockal cross in the 90th minute to further the pain of Celtic, who have lost three home games in a row in the competition for the first time and sit bottom of their group.