Joshua took a huge right hand in the sixth round and hit the canvas, but showed resilience to recover and knock out his opponent in 11 rounds.
The Brit maintained his perfect professional boxing record and challenged Fury to a meeting on the canvas immediately after his victory.
Addressing a record 90,000 crowd at Wembley Stadium, Joshua said: "I came out and I won, I didn't go into the slugfest, I came back and fought my heart out. (Tyson) Fury where you at, baby?
"I love fighting. Tyson Fury, I know he's been talking, I want to give 90,000 a chance, I just want to fight. (Evander) Holyfield, Roy Jones Jnr, you're all my inspirations. Thank you for coming out to a beautiful night of boxing."
And Fury, who beat Klitschko to become world heavyweught champion in 2015, responded on social media.
He tweeted: "Well done @anthonyfjoshua good fight, you had life & death with @klitschko & I played with the guy, let's dance."
Klitschko was graceful in defeat and said the best man won on the night.
He said: "London, I hope you enjoyed the fight, he beat me at his best, the best man won tonight.
"Two gentleman fought each other, Anthony won tonight, I (was) waiting to give but all the respect to you guys, thank you.
"I wish I could raise my hands but congratulations to Anthony."