After the fight, Joshua called out former heavyweight champion Tyson Fury, who dethroned Klitschko in 2015 but was subsequently forced to vacate his belts due to a number of issues outside the ring.
Joshua charged out with a ferocious onslaught in the fifth that sent Klitschko to the canvas, but the experienced former champion recovered well and it was the British fighter who ended the round gasping for air.
Joshua, meanwhile, now boasts an ideal 19-0 record following his win over Klitschko, with all of the 27-year-old's victories coming inside the distance. And as well as defending his IBF title, he also claimed the WBA "super" belt.
The veteran's fourth stoppage loss of his career seemed imminent, but back he fired to stagger Joshua, before dropping him heavily in the sixth.
He may not be the fighter he was at his exceptional peak, and he will decline further as an effect of such a bruising fight, but - beyond Joshua and Tyson Fury - it would be hard to argue against him beating any of the world's other leading heavyweights.
Tyson Fury responded to Anthony Joshua calling him out immediately after his victory over Wladimir Klitschko by saying "challenge accepted".
Aside from a harsh word for Klitschko, Fury called his potential fight with Joshua the "biggest fight in British boxing history".
The National debate: Anthony Joshua v Wladimir Klitschko - who comes out on top?
No reason to leave early when being the heavyweight champion is so much fun.
"I don't mind fighting him again if he wants the rematch", Joshua said at a press conference after the fight in front of 90,00 fans at Wembley Stadium. "I am right now interested".
Joshua landed the better shots in round one.
He then suffered the heaviest knockdown of the fight after a huge right hand and remained content to survive until the dramatic 11th round, when he put Klitschko down with an equally big uppercut, before stopping him on his feet after a further knockdown.
Joshua - the London 2012 Olympic gold medallist who was born in England to Nigeria-born parents - had never been beyond seven rounds, Klitschko has fought 12 rounds nine times. Anthony was better today than I. It's really sad that I didn't make it tonight. It didn't work, but all the respect to Anthony. I know I can get them, so I'm searching, and "Boom", they were just skimming his face.
WBC champion Deontay Wilder was ringside and would be a mouthwatering option, although Joshua does have an IBF mandatory obligation to Bulgaria's Kubrat Pulev.
A rematch clause is in the contract, but whether he wants to take it up is a different matter.