Kane had a penalty saved by Kasper Schmeichel before converting the rebound to send England into their first major tournament final since the 1966 World Cup.
England will play Italy in the Final at Wembley on Sunday.
A brilliant free-kick from Mikkel Damsgaard had put Denmark ahead before an own goal from Simon Kjaer pulled England level.
Neither side could find a winner in the 90 minutes but Kane scored the winner in the first half of extra-time to spark jubilant scenes at Wembley.
James Robson was at Wembley to rate the England players…
Jordan Pickford 7
Loose clearance almost let Braithwaite in. Could do nothing about Damsgaard’s stunning free kick but was commanding after that.
Kyle Walker 8
England’s best player for much of the first half – producing a brilliant recovery run to cut out Damsgaard when he was threatening run through on goal. Service to Saka could have been better.
John Stones 8
Denied Denmark chances from open play. Struggled to build from the back in the first half, such was England’s lack of control in midfield. Grew in stature as the game went on with a dominant performance in the second half and extra time.
Harry Maguire 9
Forced a brilliant save from Schmeichel with his second half header. One of England’s best users of the ball at the back and on the ground or in the air that Denmark could throw at him.
Luke Shaw 8
Not as creative as he has been in recent matches and did not get on enough set plays. But tenacious and tireless and still a threat going forward.
Declan Rice 8
Put under severe pressure by Denmark’s press. Some important interceptions and grew into the game with an awesome second half performance as he won virtually everything to keep Denmark on the back foot.
Kalvin Phillips 7
Lost the ball to Hojbjerg in a dangerous area early on and looked as uncomfortable as he has at any point in these finals. Vastly improved after the break with a far more positive performance, playing higher up the pitch.
Mason Mount 7
Vital to England’s press behind the forwards. It was a hard slog in the first half, but he was far more influential in the second when finding space down the channels to hurt Denmark.
Bukayo Saka 7
Starved of the ball for much of the first half. Burst into life once England went behind – at the heart of two golden chances, including Kjaer’s own goal to hand the Three Lions an equaliser before the break.
Raheem Sterling 9
Brilliant. Just Brilliant. Denmark could not contain his runs. It was his dart into the six-yard box that forced Kjaer to put the ball into his own net for England’s equaliser and his dribble that won the decisive penalty in extra time.
Harry Kane 8
In No10 mode here - setting up Sterling for one first half chance and cutting Denmark open with his pass to Saka for England’s equaliser. But it was his strike in extra time that booked England’s place int eh final.
Jack Grealish (Saka, 69) 7
Won a free kick and earned Wass a booking with virtually his first touch of the ball. Dragged players towards him and pulled Denmark out of shape.
Jordan Henderson (Rice, 95) 7
Brought energy and control when England needed and legs were tiring. Justified Southgate’s decision to take a chance on him after missing the end of the season through injury.
Phil Foden (Mount, 95) 6
With Denmark sitting deeper and deeper, England needed more creativity to pick the lock. Foden was just another player to torment and disrupt a team that were fading by the minute.
Kieran Trippier (Grealish, 106) 6
Came on to allow England to switch to three at the back for the final 15 minutes.
Subs: Ramsdale, Johnstone, Rashford, Mings, Coady, Sancho, James, Bellingham