The Sweden captain became the first player to score at the Friends Arena and then struck three times in the final 15 minutes after goals from Danny Welbeck and England debutant Steven Caulker had threatened to spoil the Swedish party.
The result meant that Sweden avenged their 3-2 loss to England at Euro 2012, while England coach Roy Hodgson -- who handed debuts to six players -- tasted defeat over 90 minutes for the first time in 12 games.
It also meant that England captain Steven Gerrard had to stomach a loss on the occasion of his 100th cap, as Ibrahimovic moved ahead of Henrik Larsson and into third place in the list of Sweden's all-time top scorers.
I thought it was a good performance," Hodgson told BBC radio.
"Obviously, it was disappointing to concede three goals in the last 15 minutes when we were looking so good at 2-1.
"Ibrahimovic scored a wonder goal to put the icing on cake, but for large periods of the game, I can be more than satisfied. Many players who got a chance came out with great credit."
Despite the pomp of a pre-match inauguration ceremony, a muted atmosphere and a pitch that cut up badly contributed to a low-key opening.
At 17 years and 342 days old, Raheem Sterling became England's fifth-youngest debutant of all time, but he was to play an unwitting role as Sweden broke the deadlock in the 20th minute.
The Liverpool winger was dispossessed just inside the Swedish half, allowing Martin Olsson to break forward down the hosts' left flank.
From Olsson's subsequent cross, Ibrahimovic saw a shot blocked by Caulker, but the Paris Saint-Germain hitman reacted quickly to poke the ball into the top-right corner and claim the first goal at the new stadium.
Tom Cleverley spurned a good opportunity to equalize when he could only divert Glen Johnson's drilled cross wide from six yards, while Mathias Ranegie missed an open goal at the other end after a dreadful mix-up between Joe Hart and Gary Cahill.
Four minutes later, England drew level, Ashley Young flummoxing Sebastian Larsson with a step-over and crossing for his Manchester United team-mate Welbeck to score at the back post.
Gerrard, positioned at the base of an inverted triangle in midfield, then
made his mark on the game by supplying the deep free-kick from which Caulker put England ahead.
It was a second goal in two games for the Tottenham Hotspur centre-back, who netted in Spurs' weekend defeat at Manchester City, but he was soon at work in more familiar territory, brilliantly diverting an Ibrahimovic shot over the bar.
Both sides saw chances come and go after the interval, with Sweden growing into the second half after England began on the front foot.
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The number of England debutants rose as Ryan Shawcross and Carl Jenkinson joined the fray from the bench, and Arsenal midfielder Jack Wilshere made his first international appearance after 17 months out with injury.
Crystal Palace forward Wilfried Zaha also made his debut, but the game belonged to Ibrahimovic.
He chested down Anders Svensson's lifted pass to volley home the equalizer in the 77th minute and then completed his treble with a 30-yard free-kick that eluded Hart to nestle in the bottom-right corner.
There was more to come. In stoppage time, a through-ball drew Hart racing from his line and when he miscued his headed clearance, Ibrahimovic found the net with an audacious overhead bicycle kick from 20 yards.