The hosts went into the Group J showdown knowing that they only needed to avoid defeat in order to secure second place in the standings, behind already-qualified Portugal.
That task looked in danger when they fell behind to Orri Oskarsson's opener after 17 minutes, but they were ahead before half time and had effectively wrapped up the win - and with it a spot at Euro 2024 - within 10 minutes of the start of the second half.
Lukas Haraslin was the star of the show, setting up the equaliser before adding two goals of his own in the opening stages of the second half to put Iceland into a 4-1 lead.
Juraj Kucka was the man to make it 1-1 on the half-hour mark, and just six minutes later Slovakia had completed the turnaround, with Ondrej Duda converting from the spot.
Those two goalscorers both turned provider for another quickfire double after the interval too, with Duda assisting Haraslin for his first before Kucka returned the favour to the Sparta Prague forward eight minutes later.
The four goals in the space of just 25 minutes either side of half time enabled Slovakia to start their celebrations early, despite Andri Gudjohnsen pulling one back for Iceland within one minute of coming off the bench.
The result sees Slovakia become the 11th team to confirm their place at the tournament in Germany next summer, while also consigning Luxembourg to the playoffs.
Luxembourg continued their impressive campaign with a 4-1 win over Bosnia-Herzegovina to wrap up an unlikely third place with a game to spare, keeping them in the hunt for a first-ever major tournament.
Slovakia, meanwhile, have now qualified for three successive European Championships, having never previously done so before this run.
Elsewhere in Group J, Portugal continued their 100% record in qualifying with victory over Liechtenstein courtesy of goals from Cristiano Ronaldo - his record-extending 128th international strike - and Joao Cancelo.