Tottenham Hotspur had been forced to wait and, as each one of their top-six rivals won on Boxing Day or Tuesday, it might have felt pretty tortuous. But when they finally got their festive programme under way they made it count – and how.

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Mauricio Pochettino’s team shrugged off a second-minute concession to Virgil van Dijk and a generally sloppy start to produce a show of strength, in which Dele Alli scored twice and both Mousa Dembélé and Christian Eriksen sparkled around him. Danny Rose was also excellent.

Southampton played the final 33 minutes with 10 men after Nathan Redmond’s sending-off for a foul on the goal-bound Alli and they were overwhelmed.

The visitors could even afford to miss a penalty in faintly comic circumstances – Harry Kane lifting his kick high over the crossbar after Redmond’s red-card offence – and they will remember it was in this fixture last December that they sparked their title charge with an impressive 2-0 win.

This performance was even more eye-catching. Kane had put them in front early in the second half, after Alli’s first-half equaliser, and they turned the screw once Southampton were a man down. The substitute, Son Heung-min, raced on to an Eriksen pass to drill home the third – it was the most clinical of finishes – and Alli gave the scoreline an emphatic feel with his second two minutes later.

The penalty/red-card double whammy, served up after considerable deliberation by the referee, Mike Dean, was the major talking point. It was a slick Tottenham counter-attack featuring Eriksen and Moussa Sissoko that had got Alli away and the chasing Redmond pulled at him from behind before they entered the area.

The Southampton winger then stumbled into Alli inside the box to make further, less obvious contact but it was still enough to knock the Tottenham player from his stride. Alli sent his shot wide as he went to ground. There was a dramatic pause as Dean considered his options, followed by a flourish when he flashed the red card.

Claude Puel, the Southampton manager, said the decision was “a little hard” and difficult to accept as it was simply “two players fighting for the ball” and he insisted that it had killed his team’s chances. On the other hand Pochettino was entitled to point out that Tottenham had not only been 2-1 up at the time but were well in control. And there was little doubt that Redmond had committed a last-man foul on Alli at some point.

The penalty carried shades of David Beckham’s decisive miss at Euro 2004 for England against Portugal. When Kane planted his standing foot next to the spot he inadvertently created a divot beneath the ball and, in turn, he got horribly underneath the shot.

Kane said he had noticed before the game there was a relaid patch of grass just on the penalty spot while he also tweeted that “if there’s any NFL teams looking for a kicker in the future, have a look at my game tonight”.

He could afford to joke. Alli talked of Tottenham having made a big statement and nobody could disagree.

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The only disappointment for Pochettino was in the yellow cards that Kyle Walker and Jan Vertonghen collected which will mean they are suspended for the trip to Watford on New Year’s Day. The manager is hopeful Toby Alderweireld – absent here because of a virus – will be available to return.

Pochettino saluted the character of his team and it was notable in what was a comeback win. Van Dijk had got in between Vertonghen and Rose to head home from James Ward-Prowse’s free-kick and for the first 15 minutes or so Southampton called the tune.

It was a sign of Tottenham’s irritation that Vertonghen left his hand across Jay Rodriguez’s face as if he meant it after making a clearance; the Southampton striker made no fuss and Dean took no action.

Southampton pressed hard at the outset and Redmond fizzed one low shot inches past Hugo Lloris’s far post.

Facebook Twitter Pinterest Harry Kane’s penalty kick for Tottenham against Southampton sails over the bar. Photograph: Eddie Keogh/Reuters

The game turned on Alli’s equaliser. Dembélé initiated the move with a deft pass into the inside-left channel for Sissoko and, when his cross looped up off Redmond, Alli leapt in front of Van Dijk to direct a header in off the far post.

Tottenham had further chances before the interval. Eriksen blazed high and Victor Wanyama, booed on his return to Southampton, tricked through only to be denied by José Fonte’s block.

Southampton’s loss of hustle after their flying start was difficult to fathom and, when they lost their defensive bearings shortly after the second-half restart, they found themselves behind.

It said everything that Oriol Romeu was the nearest marker to Kane on Eriksen’s corner – and he was not close enough. Fraser Forster might have done more to keep out Kane’s header.

Fonte and the Southampton substitutes, Pierre-Emile Hojbjerg and Dusan Tadic, had chances while Eriksen rattled the crossbar on 71 minutes.

But when Son and Alli showed their clinical edge, it meant Southampton had conceded four goals at home in the Premier League for the first time since 1998.