Son Heung-min has waited all season for a Tottenham goal and what do you know, he had two come along at once as he helped to disperse the storm clouds from around his club and set up a Carabao Cup quarter-final at Arsenal.

It has been a difficult period for Spurs with the news of further delays to the new stadium, a gloomy financial update, defeat by Manchester City at a half-empty Wembley on Monday night and Mauricio Pochettino’s admission that he had never felt lower at the club.

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But Son – whose season has taken in the Asia Games – got himself up and running with a couple of predatory finishes to lift hopes of a first trophy for Spurs since 2008.

Dele Alli was central to Son’s goals and after a West Ham substitute, Lucas Pérez, made it 2-1, Fernando Llorente scored his first goal of the season to smooth Spurs’ progress. It was a thumping volley from the centre-forward, after he had outmuscled Angelo Ogbonna from a Christian Eriksen corner, and the closing stages would be notable for the introduction of a 17-year-old midfielder, Oliver Skipp, for a debut.

West Ham created chances, particularly in the first half, but they lacked a threat and left with a whimper. They also had to endure the cringe‑worthy sight of two fans getting out of the home enclosures to invade the pitch. Ogbonna dealt with the first of them and immediately afterwards he lost Llorente for the clinching goal. Was his focus blurred? “When you lose, it’s not good to find excuses,” Manuel Pellegrini, West Ham’s manager, said.

Facebook Twitter Pinterest West Ham’s pitch invader wearing a Scream mask is ushered away by Arthur Masuaku. Photograph: John Patrick Fletcher/Action Plus via Getty Images

Pochettino was back in an upbeat mood, talking of his pride at the team’s resolve, and excitement at the prospect of facing Arsenal, who are managed by his good friend Unai Emery. “It’s ironic because our worst enemies are West Ham and Arsenal but I love Manuel [Pellegrini] and I love Unai,” Pochettino said. “We have very good relationships although during the 90 minutes we are going to fight.”

Pochettino had felt the squeeze from a congested programme and he retained only Davinson Sánchez from the starting lineup against City but he still fielded plenty of classy players – not least Son, the fit again Alli and Eriksen. On this evidence, Pochettino is not quite down to the bare bones in terms of squad depth.

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Son’s opening goal was marked by class and composure. He had time to size up the situation after Alli’s attempted flick had deflected off Ogbonna but he made it look easy, pinging a left-foot shot into the near top corner.

Pellegrini had made a relatively meagre five changes from the draw at Leicester on Saturday and West Ham created the chances to have led at the interval as Grady Diangana caught the eye with his pace and twinkle toes.

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The 20-year-old winger’s best moment came on 15 minutes when he tricked inside Kyle Walker-Peters and Sánchez before unfurling a glorious cross for Javier Hernández, who tried to touch back for Michail Antonio only to see Juan Foyth stretch and intercept.

Hernández looked short of confidence. His chest control let him down in the 39th minute after a Diangana free-kick ran through to him, allowing Paulo Gazzaniga to collect, while he did not get hold of his effort following another West Ham free-kick.

West Ham could also point to a clear‑cut chance for Antonio on 37 minutes. Arthur Masuaku’s through ball that took out Serge Aurier was superb but Foyth again got back to pressure Antonio and divert his shot wide.

Pellegrini took off Felipe Anderson at half-time, as he said he had planned to do – the Brazilian was dismal – and made his other two changes before the hour, one of them being Marko Arnautovic. The manager had to be bold because by then Spurs were 2-0 up.

Facebook Twitter Pinterest Christian Eriksen shoots during his first domestic start for Tottenham since 22 September. Photograph: John Patrick Fletcher/Action Plus via Getty Images

The second was a horrible moment for Masuaku, who got himself into a tangle when trying to deal with Alli’s ball forward. He ended up touching it back towards his own goal to make it a clearer opportunity for Son. The forward ran round Adrián to tap home.

Antonio had worked Paulo Gazzaniga at close quarters at the start of the second half, having got the better of the otherwise impressive Foyth, and West Ham hinted at a grandstand finish when Pérez headed home a corner from another substitute, Robert Snodgrass. It had been awarded after Foyth’s saving challenge on Arnautovic and the goal stood after a VAR consultation. Llorente, though, would restore Tottenham’s cushion.