Something is growing at Liverpool in the supremely qualified opinion of Jürgen Klopp. That judgment applies to expectations at Anfield this season, as well as the development of his refreshed, gifted and hungry team. West Ham were comfortably eclipsed as Liverpool registered a sixth successive win in all competitions with plenty to spare.

David Moyes’s visitors started brightly and deserved to head in level at the interval. Once Liverpool’s quality on the ball improved in the second half, however, West Ham were subdued and unable to contain the movement around them.

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Moyes’s long wait for a first win at Anfield goes on. Mohamed Salah gave Liverpool the lead from the penalty spot – he has now scored in 12 of his team’s last 13 home games – before second-half goals from Darwin Núñez and substitute Diogo Jota made Jarrod Bowen’s equaliser a distant consolation. Michail Antonio’s confident boast that West Ham will finish above Liverpool looks seriously misplaced on current evidence.

“It was a difficult game, like we expected,” said Klopp. “We had good moments in the first half but were not in control to deny them completely. We lost some balls we shouldn’t have lost and when they scored I was not completely surprised.

“We didn’t concede first this time but we conceded anyway, and that’s not great, and the reaction was top. That’s important because we have to be a team that’s difficult to deal with.

“We were better in possession in the second half. At 2-1 up we had our best phase of the game. We kept West Ham calm with our ball possession, which is the best way to protect. I saw a lot of good stuff today.”

West Ham were in the ascendancy and could have been two goals ahead when they presented Liverpool with an early lead. A loose header from James Ward-Prowse was intercepted by Curtis Jones and the hosts countered through Luis Díaz.

His low cross clipped Núñez’s heel and Nayef Aguerd, wrong-footed as a result, conceded the clearest of penalties with a trip on Salah. Salah picked himself up to send Alphonse Areola the wrong way from the spot.

It was a timely break for Liverpool, who were indebted to Alisson for a superb goalline save from a Tomas Soucek header and a glaring miss from Antonio, heading well wide from the Czech midfielder’s inviting cross, for not having to come from behind once again.

Dominik Szoboszlai was decorating the contest with touches of class, matching his talent with a prodigious work rate, and Liverpool pressed for the comfort of a second goal. Virgil van Dijk shot over when picked out by an Andy Robertson corner. Salah swept a first time shot wide from Díaz’s layoff. But a second remained elusive and West Ham carried a persistent threat in the form of Lucas Paquetá, Bowen and Antonio.

All three were involved in a fine and merited equaliser shortly before the break. Bowen headed Paquetá’s lofted ball into the path of Antonio, who released Vladimir Coufal down the right with a perfectly weighted pass.

The full-back threaded a cross between two Liverpool defenders and Bowen, launching himself in front of Van Dijk, steered a superb diving header in off Alisson’s right post.

Szoboszlai was unfortunate not to have a glorious assist when arcing a stunning pass behind the West Ham defence for Jones to volley home. The midfielder had just strayed offside when the Hungary international spotted an option that few others could see.

A similar routine restored Liverpool’s lead on the hour, however. This time Alexis Mac Allister floated a beautiful pass over Aguerd and Núñez, pulling off the shoulder of the last defender, guided an acrobatic volley beyond Areola. An excellent goal was reward for the centre-forward’s tireless shift plus his refusal to be affected by a glaring miss when unmarked in front of goal five minutes earlier.

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Bowen, who returned to team training after illness only on Saturday, missed a good opening early in the second half too, heading a Ward-Prowse free-kick straight at the Liverpool goalkeeper from inside the six-yard area. It was a costly miss as West Ham struggled to match the home side’s intensity or composure in possession thereafter.

Liverpool added the third their second-half dominance deserved late on from a Robertson corner. Van Dijk soared above the West Ham defence to head the delivery back across goal and Jota, perfectly placed five minutes after his introduction from the bench, swept in from close range.

Moyes reflected: “We did a lot of good things today but I come away from here saying the same things too often. I’m sick of saying it. We ran both Manchester City and Liverpool close in the last two games but we gave three goals away in the second half against City and two away in the second half today.

“I have to find a way to put in the same energy levels in the second half as the first, but overall it has been a really good start to the season by us.”