A post-war record 90,000 fans at the national stadium were treated to a see-saw contest, Joshua dropping the former unified world champion in the fifth round, before being put down - for the first time in his 19-fight career - in the sixth.
Both men looked in danger of being stopped over two rounds which will live long in the memory, before Joshua delivered the clinical blows in the 11th.
The Briton landed a brutal uppercut which gave him the platform to send Klitschko down with an immediate flurry.
The Ukrainian stood, was dropped by a left hook again moments later, then stood again, only for referee David Fields to save him when pinned against the ropes.
Joshua threw his arms into the air as roars rolled down the Wembley tiers. He had come through his stiffest test to date, while Klitschko too had answered questions, showing he remains competitive at world level, even at 41.
He will rue not finishing the job when he floored Joshua and appeared tantalisingly close to reclaiming two of the titles he lost in his last fight against Tyson Fury in 2015.
At the time of the stoppage, Joshua was up 96-93 and 95-93 on two cards, with Klitschko leading 95-93 on the other.
After lengthy ring walks, fireworks, deafening noise and a rousing British national anthem, an opening six minutes of skill over power played out.
Klitschko flicked his jab, his left hand waving in front of Joshua's face. Feints from both men stood out as they hunted openings.
A solid right hand early in the fourth caught Joshua, who responded with the same shot later in the round, nodding at his rival on the bell.
And then came the power punching in a pulsating fifth round. Joshua shook his man before sending him down face first with a flurry of punches.
The crowd sensed blood, urging the Briton forward, only for Klitschko - cut above his left eye - to show the heart which has made him the most dominant heavyweight of the 21st century.
He pinned Joshua to the ropes, turning the round on its head, and then came out in the sixth to administer an overhand right which crumpled the champion. In uncharted waters, Joshua answered the count and faced a stiff examination before the bell.
This was now an epic, living up to the hype, heavyweight boxing at its most captivating.
The pace dropped, it had to, though any fear of Joshua's career-high weight slowing him down was unceremoniously put to bed in the 11th as his power unified the titles.
The sweat flew from Klitschko's head as the telling uppercut landed crushingly, before Joshua hunted down his opponent to seal a historic win.