Spurs began woefully in their last-16 first leg in Turin on Tuesday as Gonzalo Higuain scored a volley and then a penalty within the first nine minutes of the match.
However, the Argentine missed a spot kick on the stroke of half time, after Kane had halved the deficit, and Christian Eriksen then struck midway through the second half to level proceedings.
"We could have [capitulated], away from home in the Champions League. It showed character," the England striker told BT Sport after the game.
"It was an excellent performance after that and we take two away goals to Wembley. Great result. I should have scored the header but you have to be ready for the next one and I finished it off."
Spurs meet Juve in the second leg at England's national stadium on Wednesday, March 7.