Alexander Isak and Viktor Gyökeres each had a goal and an assist, and Mattias Svanberg also scored for Sweden in Guadalupe on Sunday night (Monday AEST).
The Swedes, who entered the tournament ranked 39th in the world, had missed the 2022 World Cup after reaching the quarter-finals eight years ago in Russia.
Omar Rekik scored for 45th-ranked Tunisia, who are playing in their seventh World Cup but have never advanced beyond the group stage.
The higher-ranked teams in the group, the Netherlands and Japan, played out a 2-2 draw earlier on Sunday at Arlington, Texas.
Ayari bookended the scoring for the Swedes at Estadio BBVA with long-range goals in the seventh minute and in second-half stoppage time.
He was muted in his celebrations, raising his hands out of respect for the country where his father was born.
Isak scored his 18th international goal in the 30th minute when Gyokeres found the Liverpool striker - who was sidelined much of last season with a broken leg - on a counter-attack.
Isak cut inside and fired a low shot past the hands of Tunisian goalkeeper Mouhib Chamakh, who struggled all night.
After Rekik stalled Sweden's momentum with his goal in the 43rd minute, Isak and Gyokeres put Sweden back in control.
Isak stole the ball and found Gyokeres for a one-on-one against Chamakh that the Arsenal striker won easily for his 16th goal in his past 15 appearances for Sweden and 21st overall.
Svanberg scored in the 84th minute, a goal that was awarded after a lengthy video review.
The Swedes dominated the match and had more shots on goal than Tunisia, leading the count 13-6, including a 7-2 advantage in shots on target.
Sweden will play the Netherlands at Houston on Saturday, while Tunisia will remain in Monterrey and face Japan the same day.
- with Reuters
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