England were once again poor, looking lethargic and unable to create clear openings, but a moment of magic from Bellingham, who fired home with a bicycle kick, kept them alive before Kane's header set up a clash with Switzerland in Duesseldorf on Saturday.
Slovakia started far sharper than England and gave their opponents several warnings before Ivan Schranz placed a deft finish past goalkeeper Jordan Pickford in the 25th minute.
England, who had come back to win their last three Euro games in which they conceded first, including the last edition's semi-final, thought they were level shortly after the break but Phil Foden was clearly offside before tapping home.
Kane uncharacteristically headed a Foden free kick wide and Declan Rice hit the post as Slovakia were pinned deep in their own half late on, though England seemed to lack belief as their fans grew ever more frustrated.
But Bellingham, a day after his 21st birthday, equalised with an incredible effort to take the game to extra time with England's first shot on target.
"What an unbelievable goal, he's kept our tournament alive," Kane told ITV, describing Bellingham's goal as "one of the best in our country's history".
"You're 30 seconds away from going home, feeling like you've let your nation down, and one kick of the ball and everything's great," Bellingham said. "It's a feeling I don't want to be in but, when it comes, it's a great feeling."
England took less than a minute of extra time to take the lead when Kane headed past the despairing Martin Dubravka from close range to complete a remarkable turnaround.
Slovakia had faded as the game drew on and went closest to an equaliser of their own just before halftime in extra time, when Peter Pekarik kneed a dangerous cross over the bar.
England repelled a late wave of balls into the box to secure a fortunate victory and their progress to the next round, when they will need to be far better against the Swiss who outplayed defending champions Italy in the last-16.
The win will not protect England manager Gareth Southgate from criticism, however, after he took 66 minutes to make a substitution despite his side looking bereft of ideas.
SLOVAKIAN REGRETS
Slovakia had deserved the lead when Schranz scored his third goal of Euro 2024, making him joint top scorer with Georgia's Georges Mikautadze and Germany's Jamal Musiala.
A simple ball forward was nodded down by Juraj Kucka to David Strelec, who had time and space to pass the ball to Schranz who placed it beyond Pickford.
It could have been far worse for England as defender Marc Guehi, on a yellow card which means he is suspended for the next round, clipped his heels but Schranz stayed up to slot home.
England should have been level in the 50th minute when Kieran Trippier was played into space but, rather than shooting, passed inside to Foden who had needlessly strayed ahead of the ball.
When Kane, largely anonymous throughout the contest, spurned a good opportunity before Rice hit the woodwork, it seemed as though England's luck was out.
But they forced extra time when a long throw into the box was nodded on by Guehi before Bellingham's brilliance left Dubravka with no chance.
Kane then met substitute Ivan Toney's flick-on with a powerful header to give England a lead they rarely looked like squandering and keep Euro 2020's runners-up in the tournament.
"We played on par basically with England, we almost qualified," Slovakia coach Francesco Calzona said. "We were very nearly there, unfortunately things went the way they did."
Slovakia's goalscorer Schranz said: "It could have been beautiful, great, but we lost out at the last moment. But we certainly have nothing to be angry about.
"We'll deal with it, but maybe we'll regret it for the rest of our lives."
