Sweden beat Italy 1-0 on aggregate to reach Russian Federation next summer, but manager Janne Andersson was bombarded with questions about a potential Ibrahimovic return post-match.
Four-times champions Italy, who had not missed the World Cup since the 1958 finals in Sweden, penned the visitors into their own half for most of the game and created a hatful of chances but could not find a way past Robin Olsen in the Sweden goal.
"Guys, we didn't lose the World Cup today [Monday]. I have not yet spoken to the president", said Ventura. I am a bit moved but obviously very happy with the result.
Italian fans were struggling to come to terms with the prospect of a World Cup without their national team.
"I said to the lads in the dressing room that if we got to the World Cup they could shave [my hair] off", he said.
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"For my part, this is the biggest thing that has happened to me and for those of us that are older this is probably the last chance to play at a World Cup so to succeed in those circumstances is an unbelievable joy", the 32-year-old said. Let us return to being the Italy that the entire world envied!
Italy's players collapsed onto the pitch in despair as the final whistle went and even veterans such as Giorgio Chiellini were inconsolable.
Ventura had made changes from Friday's defeat with Brazil-born Jorginho making his Italy debut and Alessandro Florenzi and Manolo Gabbiadini also handed starts, although Napoli's in-form winger Lorenzo Insigne was again left on the bench.
Failure to qualify has heaped pressure on Italy boss Gian Piero Ventura, with the sport daily Corriere dello Sport calling for him to step down, while the headline in La Gazzetta dello Sport read "Italy, this is the apocalypse".
"It is an intolerable football shame, an indelible stain", the newspaper said.