Chris Waddle was following the match for BBC Sport, and while the former Tottenham star was impressed with the performance of the 30-year-old, he has criticised the midfielder for not scoring enough goals.
Juventus' second goal came in the 8th minute of the match.
Douglas Costa's pace drew an ill-judged foul from Serge Aurier inside the box and the last kick of the half was another chance for Higuain to score from the spot.
"I feel so proud", Tottenham coach Mauricio Pochettino said. "We showed great character against a team that played two finals in three years, with unbelievable experience on the pitch". We are facing Tottenham in the Champions League, a really physical and technical team.
"When you see a performance like that, to re-group and to comeback and play as they did, it was a controlled performance. They are a very physical and technical team".
"I think people have lost a bit of objectivity", said Allegri. Juve play to win, but they are not the favourites.
"Now it'll be like a final in London".
MF Eric Dier, 7 - Outshone by other teammates but did his job well, repeatedly altering the shape of the team by moving between the right side of defence and central midfield, depending on which team had the ball. We scored twice against a team that is always hard to break down.
MF Mousa Dembele, 9 - Gave away the free kick that produced the opening goal but was otherwise superb.
Kane's long-range effort was bundled away by Buffon but the visitors soon won a free kick, which was sacked into the bottom corner by Eriksen via the hand of Buffon who was slow to get down to his right. In the last few weeks, the team are growing and have stepped up. "You can't reach the final every year", Allegri said.
Juve had only conceded one goal in their previous 16 games.
Higuain nearly had a hat-trick, picking up a Pjanic pass to outfox two Spurs defenders only to shoot just wide.
Tottenham were wobbling but responded to dominate for long periods and Kane halved the deficit in the 35th minute with his ninth goal in nine appearances in the competition.
Medhi Benatia gifted Eriksen a chance and Kane went sprawling to the ground in the area but German referee Felix Brych was having none of it.
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.
Pochettino and Spurs were left with the dilemma of whether to stick or twist, but Juventus kept on looking for a third goal and it nearly came when Bernardeschi's deflected effort almost crept past Lloris at his near post.