The 22-year-old Serb, a £58m marking from Fiorentina last month, ran onto Danilo's long ball before a turn and cunning completion following 32 seconds.
Villarreal's Giovani lo Celso cut a shot off the outside of the post.
The leveler came when Dani Parejo ghosted in to complete from Etienne Capoue's elevated ball into the crate.
Juve were lucky to end with 11 men as Adrien Rabiot was just reserved for a risky foul on Samuel Chukwueze.
"It was tremendous to score on my introduction," said Vlahovic, whose main other European experience was as a 16-year-old for Partizan Belgrade in Europa League qualifying. "It was exceptionally enthusiastic.
"Notwithstanding, I can't be 100 percent fulfilled in light of the fact that we have not dominated the match. By and by, I have as of now failed to remember this game and I'm centered around the following."
With the away objective rule rejected for this present year the tie is level going into the second leg in Turin in three weeks.
Massimiliano Allegri's Juve are unbeaten in an hour and a half since November yet they were made to pay for their cynicism at Estadio de la Ceramica.
They moved to a back five in the wake of scoring that first-minute objective and afterward turned off as Parejo was plain to fire in a leveler.
A Vlahovic shot from fresh in the 85th moment was their main other shot on track.
To exacerbate the they lost midfielder Weston McKennie to a lower leg injury in the last part.
Villarreal, who have never lost a home game in the Champions League knockout stages, pushed for a victor however certainly missed harmed Spain striker Gerard Moreno.
Arnaut Danjuma, who has assigned front and center and scored a full go-around at Granada on Saturday, had a sharp backheel shot saved in the principal half.
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