Spurs needed at least a point to keep their hopes of reaching the last 16 alive.
Spurs needed at least a point to keep their hopes of reaching the last 16 alive and Kane brought them level with a penalty shortly after Djibril Sidibe had put Monaco in front at the Stade Louis II.
However, Thomas Lemar ended a whirlwind spell by restoring Monaco's lead in the 53rd minute and the hosts -- for whom Radamel Falcao had an early penalty saved -- held on to claim the win that secures their place in the knockout stages.
Monaco are also guaranteed to progress as Group E winners, while Bayer Leverkusen are through too after drawing 1-1 away to CSKA Moscow in Russia earlier in the day.
"It is a beautiful victory. We are happy and let's hope this continues," Sidibe told beIN Sports of a Monaco side who are through to the last 16 for the second time in three seasons.
"Our objective was qualification and we are first, which is important. We will savour this before we start looking forward."
Meanwhile, Tottenham will go into their final match against CSKA at Wembley with only a berth in the Europa League after Christmas up for grabs.
It is a deeply disappointing outcome for Mauricio Pochettino's men, who have now managed just one win in their last nine matches in all competitions and have lost three of their five games in Europe.
"After we lost the first goal we came back with the penalty but it is difficult to understand how we conceded the (second) goal," Pochettino said.
"If you want to go to the next level in the Champions League you can't concede goals like that."
The Premier League side had showed their intent as Kane, Dele Alli and Mousa Dembele all started for despite having been cited as doubts before the game by Pochettino.
Meanwhile, Monaco coach Leonardo Jardim named the attacking line-up that has made his team the most prolific in Europe's leading leagues this season.
However, Tottenham missed a glorious chance to go ahead inside six minutes when Alli's defence-splitting pass sent Song heung-Min racing clear.
The Korean found himself through one-on-one with Danijel Subasic but was pushed wide by the Croatian goalkeeper and the opportunity was gone.
Four minutes after that Monaco won a penalty. Fabinho went down in spectacular fashion under an Eric Dier challenge and Dutch referee Bjorn Kuipers pointed to the spot, only for Hugo Lloris to dive low to his right and save Falcao's kick.
The home side did not let that miss get to them, instead continuing to look the more dangerous side but it took them until three minutes into the second half to get the breakthrough.
Bernardo Silva found left-back Benjamin Mendy, who took his time before holding up a cross for fellow full-back Sidibe -- a summer target for Arsenal -- to head home from six yards.
The response from Tottenham was swift, as they won a penalty just three minutes later when Kamil Glik was adjudged to have fouled Alli in the area.
Kane stepped up to score from 12 yards and break his Champions League duck.
And yet the riposte from Monaco was quicker still as they went back in front straight from the restart.
Three passes and then a cross from the right saw the ball fall to Lemar, who took a touch before scoring with a composed low finish across Lloris.
Spurs' attempts to draw level were in vain, and only a stunning reflex save by Lloris stopped Glik from scoring after the Pole connected with a Lemar free-kick midway through the second half.
The former Nice goalkeeper also produced a good stop to deny Falcao, but Tottenham were out.