The tie seems already settled ahead of the second-leg at the Etihad on March 7.
Asked if he feared his side were about to sink without trace after Higuain's double strike, Kane told BT Sport: "We could have done, away from home in the Champions League". "We look back now, two away goals we're taking to Wembley, it's a great result".
"In any football match, you can have ups and downs".
Moments earlier, however, Dele Alli had only been denied by a last-ditch Bernardeschi challenge and Tottenham's dominance finally paid off when the midfielder slipped Kane in behind. Dembele ran the game, brushing off Juventus players who tried to get close to him and dribbling out of tight spots, his passes forward was crisp and incisive, and he won the ball back to restart moves.
Juventus coach Massimiliano Allegri went for an attacking lineup, with Federico Bernardeschi, Douglas Costa and Mario Mandzukic supporting Higuain.
Tottenham arrived full of confidence after winning their group containing last year's winners Real Madrid, but they suffered a terrible opening 10 minutes. Pjanic himself took the free kick and picked out Higuain, who beat the offside trap to volley into the far corner.
The Tottenham striker delivered on the big stage, scoring in the Champions League last 16 when his team needed him most.
And Kane's latest goal helped make Tottenham slight favourites to reach the quarter-finals.
But from there on, Spurs dominated the game as Max Allegri's men begun to sit deeper and deeper inside their own half.
Higuain nearly had a hat-trick, picking up a Pjanic pass to outfox two Spurs defenders only to shoot just wide.
However, the Premier Legaue side fought back, with Kane rounding Gigi Buffon before firing home on 35 minutes.
Spurs pushed for the equaliser but Juventus nearly extended their lead on the stroke of halftime when Serge Aurier brought down Douglas Costa in the area.
He could not complete his hat-trick, however, as the ball thundered against the bar and the players headed for the dressing room.
Juve, who had conceded only once in their previous 16 games in all competitions, will rue Higuain's missed penalty on the stroke of halftime but could have few complaints after being shaken to the core by a vibrant Spurs side.