Celtic attacker Elyounoussi went some way to easing the growing pressure on boss Neil Lennon with his brace in the space of four minutes against the in-form hosts, who are now unbeaten in 10 games this season.
Scott Bain kept out Jonathan David's penalty and Sven Botman hit the crossbar before the interval in what was looking like being a frustrating day for Lille.
But a first victory away at French opposition in nine attempts since 1966 ultimately eluded Lennon's men as Zeki Celik and Jonathan Ikone struck in the second half to snatch a point for the Ligue 1 club.
Elyounoussi pulled the strings for Celtic from the early stages, setting up Albian Ajeti for a goal that was disallowed for offside and another chance that his team-mate skied over.
But the Norway international took matters into his own hands by firing the visitors ahead with a fine curling effort from 25 yards that left Mike Maignan stranded.
The finish for Elyounoussi's second was just as crisp as he guided a first-time shot away from Maignan into the bottom-left corner after being spotted by Jeremie Frimpong.
David was denied by Bain from the penalty spot after being felled by Shane Duffy and Botman clipped the top of the crossbar with a shot as Lille pushed for a way back into the game.
And it was given a lifeline through unmarked Celik's finish at the back post, the full-back bundling the ball past Bain after a corner was flicked into his path.
Celtic was under siege and a second goal followed eight minutes later as Yusuf Yazici's mishit shot fell kindly for Ikone to spin and fire past Bain, ensuring the points were shared.