Gareth Southgate's side took the lead at Hampden Park through Alex Oxlade-Chamberlain, but a couple of Leigh Griffiths free kicks in quick succession saw Scotland turn the game on its head in the final minutes of normal time.
However, Kane - wearing the captain's armband on the night - hit back in stoppage time to rescue a draw for the visitors and keep them two points clear at the top of World Cup qualifying Group F.
"[The character shown] was fantastic. I think going 2-1 down in the manner that we did, a lot of heads could have dropped or people could have been sulking or so forth. But everyone stood up to it. They dug deep and found a goal out of nothing, really, so that's what it's about. It is about standing up and being counted and we did that today," he told reporters.
"Obviously I was wearing the armband today but I think we've got a lot of leaders in the team. It was great for me personally to lead the team out, a very, very proud moment and to get that goal at the end made it even more special. I think being part of a leader is standing up when it matters and to get that goal at the end was important.
"Scotland scoring two goals late on and you look at four minutes left, so to get anything from the game is always a point gained. We come away fairly happy with it and in a good position to qualify for the World Cup. I don't think it was tougher than what we expected because we always expect to have tough games, especially against your rivals.
"We knew it was going to be tough away from home, international qualifier, Scotland are doing everything to stay in the World Cup. We knew it was going to be tough and credit to them for standing there and getting the two goals, but it was nice to obviously not lose the game."
England, who have not lost a qualifying match since October 2009, will now turn their attention to Tuesday's friendly against France in Paris.