Mauricio Pochettino wants to scream when he is told that he must win a trophy to validate his work at Tottenham. The manager wants supporters to see the bigger picture; how he has the club punching above their weight when it comes to things like balance sheets.

Pochettino wants to win the Carabao Cup purely for the thrill of winning and his team put their noses in front at the halfway stage of the semi-final before the Stamford Bridge second leg, which has been scheduled for Thursday 24 January.

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It was far from the most free-flowing performance of a season that promises so much for them and Pochettino was left to enjoy the grit they showed to keep Chelsea out. The visitors were the team that pushed and they created the chances to have taken reward. But Maurizio Sarri – who is also without a trophy from his managerial career – could lament Chelsea’s lack of cutting edge.

Sarri gave a start on the right wing to Callum Hudson-Odoi, the 18-year-old who is the subject of a £35m offer from Bayern Munich, and he had some nice moments, including a deflected cross in first-half stoppage time that Paulo Gazzaniga clawed against the post and to safety.

Álvaro Morata had been ruled out with a slight hamstring injury and Sarri preferred Eden Hazard to the fit-again Olivier Giroud in the central attacking role. Hazard was dangerous, as always, but he could not find a way to goal. Sarri sent on Pedro for Willian and Giroud for Hudson-Odoi but, whichever way he sliced it, Chelsea were blunt where it mattered the most.

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There was bad finishing, particularly from the centre-half Andreas Christensen when gloriously placed on a set piece on 58 minutes, but also bad luck. N’Golo Kanté was denied in the 40th minute by the same post that would thwart Hudson-Odoi.

It was Harry Kane, inevitably, who made the difference and when he banged home from the penalty spot midway through the first half it was his 160th goal in Spurs colours, lifting him above Cliff Jones into fourth place on the all-time list. A reminder: Kane is only 25. The execution was nerveless and clinical.

Kane had won the kick when he fastened on to Toby Alderweireld’s long ball and touched decisively past the outrushing Kepa Arrizabalaga before being clattered by him. It was a clear foul. Yet there was confusion because the assistant referee raised his flag for offside against Kane. It was extremely tight and so we went to VAR.

Facebook Twitter Pinterest Harry Kane was onside by a whisker before winning and converting the penalty that ensued. Photograph: Sky Sports

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It took an age to clear up but Kane was deemed to have timed his run to perfection. There could surely be no arguing with the technology, or could there? The Chelsea captain, César Azpilicueta, led the protests to the referee, Michael Oliver, during the interval while Sarri would later insist that, having seen the replays on Chelsea’s feed, Kane looked offside. It was an argument that ran and ran.

Wembley had been the scene of Chelsea nightmares in late November when they did not turn up for the Premier League fixture against Spurs. The 3-1 scoreline was a mercy. When Sarri reflected recently on his first season in English football, he mentioned the losses to Wolves and Leicester as his regrets. The Spurs mauling? “I don’t talk about that,” he said.

Pochettino employed the same system, with Dele Alli in a roving role behind Son Heung-min and Kane; partly to construct a creative platform for him, partly to have him stifle Jorginho when Chelsea had possession. That had been the basis of the league victory. It did not work here.

Alli released Son early on with a ball over the top that Christensen just about dealt with, while Kane sent an overhead kick at Arrizabalaga but Chelsea, who flickered at the other end, could feel that they made a solid start. After the goal they were the better team.

Hudson-Odoi had worked Gazzaniga with a low shot on seven minutes and thought he had his moment when he crossed and watched the ball loop up off Danny Rose. Gazzaniga did well to backtrack and push against the post.

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Spurs tried to build from the back and that involved risking the ball as Sarri’s players pressed. There were murmurs of exasperation from the home support. Kane extended Arrizabalaga on 52 minutes but it was Chelsea who showed coherence and came to pen Spurs in. They were the team in control.

Hazard, who turned sharply and probed throughout, worked Gazzaniga on two occasions from distance while Kanté also forced him into a save on 55 minutes. The moment when Chelsea’s hearts truly skipped came shortly afterwards. Ross Barkley flicked on a corner and there was Christensen, all alone at the far post. He scuffed his shot horribly.