Graham Potter’s men took the lead in Guadalupe in the seventh minute courtesy of Brighton midfielder Yasin Ayari’s thunderbolt from outside the box, following a mix-up at the back.
The celebrations of Ayari, who is of Moroccan and Tunisian descent, were muted despite his fine finish.
Sweden doubled their lead on half an hour after a rapid break freed Alexander Isak on the left.
The Liverpool forward raced forward and cut inside before unleashing a shot, which goalkeeper Mouhib Chamakh failed to keep out, even though he got a hand to the ball.
Tunisia did not concede a single goal in qualifying, becoming the first side to achieve that feat, subsequently matched by Ivory Coast and England.
Sweden threatened to overwhelm their opponents but the match changed complexion minutes before half-time when Omar Rekik headed home Hannibal Mejbri’s teasing cross.
However, the Scandinavian nation restored their two-goal cushion in the 59th minute after another defensive calamity for Tunisia.
Midfielder Ellyes Skhiri was caught in possession on the edge of the box by Isak, who fed Viktor Gyokeres and the Arsenal man fired home.
Substitute Mattias Svanberg made it 4-1 late on after VAR ruled he was onside.
And there was still time for another stunning goal from Ayari from outside the penalty box.
Sweden reached the quarter-finals of the 2018 World Cup in Russia but did not qualify for the tournament in Qatar four years later.
Tunisia were the first African team to win a World Cup match when they beat Mexico in 1978, but they have never progressed beyond the group stages. (AFP)
