Celtic trailed to Manuel Lazzari's 40th-minute strike against the run of play, but Ryan Christie levelled 23 minutes from time and Jullien snatched all three points with his free header at the last.
The Scottish champion lacked a cutting edge for large parts of the game and could have fallen two goals behind as Joaquin Correa hit the post two minutes before Lazio was pegged back.
The dramatic turnaround ensures Neil Lennon's unbeaten side remain top of Group E, four points better off than third-placed Lazio and a point in front of Cluj, who beat nine-man Rennes 1-0.
Lazzari rounded off a swift counter late in the first half to put Lazio in front, the one-cap Italy international cutting inside from the right and firing past Fraser Forster at his near post.
Despite seeing far less of the ball in the attacking third, Lazio should have doubled their lead 65 minutes in, only for Correa to pick out the post after being played in by Sergej Milinkovic-Savic.
That one-on-one miss proved costly as Odsonne Edouard dribbled into the box and pulled the ball back for Christie, who curled his shot past Thomas Strakosha to level the scores.
Forster produced a big save down low to deny Marco Parolo 10 minutes from time, and it was Celtic who claimed a huge victory courtesy of Jullien's back-post header after being left alone at a corner.
There was nearly time for another late twist in Glasgow, but Forster somehow got across to keep out Danilo Cataldi's drive five minutes into added time.