Spurs paid the price for another passive start that allowed Milan to settle into their task – the misery complete when defender Cristian Romero was sent off late on after receiving a second yellow card for a wild challenge on Theo Hernandez.
A late Kane header was brilliantly kept out by Maignan before Divock Origi struck the post for the visitors as Spurs went out tamely, their season now resting on the battle for a top-four place in the Premier League.
Spurs simply not good enough
The lack of urgency that has marked so much of their season has never been more stark than in the first 45 minutes when Spurs played as if they were barely aware they needed to score to keep their ambitions alive.
Manager Antonio Conte returned to the technical area after recuperating from gallbladder surgery but this did not spare him the wrath of the Spurs support mystified by such a conservative approach.
He was not helped by Romero’s red card for a late, reckless tackle on Hernandez, having already received a first-half caution for a clumsy challenge to prevent Rafael Leao breaking clear.
Kane almost rescued Spurs and sent the game into extra time with that late header, but for most of the night he was starved of meaningful service in a side largely devoid of creativity.
What this means for Conte’s long-term future remains to be seen, with growing uncertainty surrounding his position and his contract up at the end of the season.
Spurs now have only that top-four place to fight for, and on this form you would not back them to achieve it. They have scrambled through much of this season despite indifferent form but there was to be no reprieve here – and they did not deserve one.
Tomori top class for Milan
Fikayo Tomori has slipped out of the England reckoning, but his calm and measured performance was a timely reminder of the qualities he can offer should manager Gareth Southgate require them.
The 25-year-old former Chelsea defender has matured in Serie A and was a rock in the Milan defence here – strong in the air, showing impeccable timing with his tackles and calm on the ground.
Milan were also well-served by keeper Maignan, faultless throughout and producing a moment of brilliance to keep out Kane in the closing seconds.