What looked to be a 0-0 draw morphed into a 2-2 thriller in the closing half hour, as substitute Alex Oxlade-Chamberlain got the scoring started in the 70th minute.
Leigh Griffiths had turned the game on its head, netting two stunning free-kicks on 87 and 90 minutes to give the Scots a 2-1 lead.
But Tottenham superstar Kane, England's captain on the day, popped up with a 93rd-minute victor to make sure the points were shared.
The remarkable match means England remain top of Group F on 14 points from six matches, while Scotland lie fourth with eight points. I got carried away a bit, and when the boys grabbed me I just wanted everyone to savour the moment because these moments don't come around too often.
Scotland appeared to have secured a first victory over its fierce rival since 1999, but the Premier League leading scorer for the last two seasons came to his country's rescue.
He curled a 25-yard free kick past Joe Hart into the top corner in the 87th, and repeated the trick just three minutes later as he bent the ball into the opposite side of the goal.
But Kane stunned the Tartan Army in the second minute of stoppage time with an equaliser in what was his first game as England captain. Leigh Griffiths missed the Monday training but assistant manager Mark McGhee has since confirmed the player's availability for the game, stating that the Celtic striker has returned to training and is fit to face England on Saturday. "One point is not what we wanted but nothing in football surprises me", he told BBC Radio 5 live.
"People have said that there is a lack of leaders in this team but I don't think that is the case", added Southgate.
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The results showed Conservatives winning 318 seats in a House of Commons where 326 seats are needed for the barest of majorities. Page said Corbyn , a lifelong left-wing activist who has spent decades speaking to crowds, was underestimated as a campaigner.
"It is a special day". As a manager that was hard work.
"But the "intae them" theory died with the pass back rule and the tackle from behind, better pitches".
"That was beyond the call of duty at times because nobody, nobody in here or any supporter knows how hard it is to play against these players on a day like that".
"I am disappointed but it's important we have that never-say-die attitude".
"It is annoying for them -- I can't be annoyed".
Other issues to arise from the game were Chris Smalling being selected ahead of John Stones in a back four.
Oxlade-Chamberlain nearly turned provider moments later when he clipped the ball to the back post only for Lallana to nod over with space around him.
Livermore hit the post five minutes into the half after his shot took a heavy deflection off Robertson and Scotland, with James McArthur on for the injured James Morrison, survived some nervous seconds before the ball was finally safe.
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