Defeat for Chelsea shattered their hopes of qualifying for next season’s Champions League as Tottenham opened up an eight-point lead over their closest challengers for fourth place.
Ending their long wait for a win at their London rivals is just the latest sign of the progress made under Pochettino.
And it was even more impressive as they came from behind as a huge error from Hugo Lloris gifted Alvaro Morata the opening goal when the Tottenham goalkeeper misjudged a cross in the 30th minute.
Christan Eriksen’s sublime long-range strike drew Tottenham level in first-half stoppage time.
Alli took centre stage to end a barren run with two smart finishes in the 62nd and 66th minutes.
Tottenham could even afford the luxury of doing the damage without Harry Kane, who made his return from a three-week layoff with ankle ligament damage 16 minutes from time.
It was a sweet moment for Alli, who has endured criticism for some erratic performances this season.
“He’s a great talent and I have no doubts about him. Young players need room to breath,” Pochettino said.
“There’s pressure on their shoulders. But he has nothing to show to me or Tottenham.
“It’s more about the expectation being so high. I’m so happy for him.”
After a long run of poor results in away fixtures against their top four rivals, Tottenham finally proved they can win when the stakes are highest.
Pochettino, yet to win a trophy with Tottenham, said: “After 28 years it is important to win here. It’s a happy day for everyone – fans, players. We feel very proud.”
Chelsea manager Antonio Conte admitted Eriksen’s equaliser was a hammer blow after his side had dominated for much of the first half.
“The goal at the end of the first half it wasn’t good for us and our confidence. We deserved to go in 2-0 up but football is this and Eriksen scored a fantastic goal,” he said.
“Second half we tried to score again but Tottenham have been more clinical than us. In this season we are not so clinical and then we lost the game.”
– Conte rues Chelsea flop –
Failing to qualify for the Champions League could trigger Conte’s departure at the end of the season after he spent much of the campaign feuding with the club’s hierarchy over their failure to back his transfer plans.
But Conte wouldn’t concede Chelsea are out of the race for the top four.
“For sure we lost a great chance today to reduce the gap with Tottenham,” he said.
“There are seven games to go and we have to do our best until the end. At the end of the season you finish in the position you deserve.”
At the Emirates Stadium, Arsenal warmed up for Thursday’s crucial Europa League quarter-final against CSKA Moscow with three goals in the final 15 minutes against second bottom Stoke.
Pierre-Emerick Aubameyang scored the first two, including a penalty opener.
Alexandre Lacazette netted the third, also from the spot, after Aubameyang passed up the chance of a hat-trick to give his out-of-form team-mate a welcome confidence boost.
Asked if he was surprised Aubameyang did not take the second penalty, Arsenal boss Arsene Wenger said: “No, because I know they have a good understanding.
“I like that because it can only make the team stronger. That shows well the state of our society because when people are generous we are surprised.”
Frustrated by the first penalty decision, Stoke manager Paul Lambert called for the Premier League to introduce the Video Assistant Referee system that has so far only been used in cup competitions in England.
“I’ve seen VAR work. I saw it in Germany and it worked seamlessly, absolutely no problem at all,” he said.
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