When Liverpool were at the Vitality Stadium early in November, Storm Ciarán raged and they won to advance in the Carabao Cup. There was another weather warning on Sunday, with Storm Isha incoming, and another powerful message. Liverpool are back from their winter break and they have their eyes on the Premier League title.

Once again, Storm Darwin blew. In the cup tie, it was Darwin Núñez who scored the winner and it was his goal at the end of the slickest of moves at the start of the second half that tilted this game in his team’s favour.

The Uruguayan was very good in the second half, all power and threat, and he was involved in the second goal that made the points safe, Liverpool extending their lead at the top of the table to five points over Manchester City, albeit having played an extra game.

Diogo Jota applied the finish after Núñez had challenged Illia Zabarnyi and the substitute Cody Gakpo played the killer pass. After Jota scored again for 3-0, lashing home after an initial miskick, it was Núñez who had the last word. Joe Gomez’s cross was a beauty and what about the finish, Núñez steering into the far corner on the volley with the outside of his right boot.

Liverpool are fighting on four fronts; they have the second leg of their Carabao Cup semi-final at Fulham on Wednesday night, 2-1 up from the first. It is the league title that they want most of all and the way they changed up aggressively through the gears after a slow-burn first half demonstrated the confidence that is running through them.

Bournemouth are a different proposition now compared to early November, having made the needed adjustments under Andoni Iraola. There was one pre-match statistic that had them as the form team in the division over the past eight matches.

They kept Liverpool at arm’s length during the first period, putting bodies on the line when required; Neto was not stretched in goal. When Harvey Elliott slipped a ball up the inside right for Luis Díaz, his cross was blocked before coming off him and going out. It summed up the first half for Klopp’s team. Núñez got an attempted volley all wrong at the far post. “You’re just a shit Andy Carroll,” chanted the home crowd. The Liverpool support enjoyed singing that back at the end of the match.

A major talking point came on 36 minutes when Justin Kluivert stretched into a tackle on Díaz, eyes on the ball, but he got nothing of it and crashed into his opponent’s lower leg. It looked bad on the replays and it seemed as though the VAR could intervene. He did not, with Kluivert escaping without yellow or red. Liverpool had played on for a while as Díaz writhed in agony so perhaps they did not think it was so bad in the moment. Players have been dismissed this season for similar challenges.

Bournemouth stirred as an attacking proposition just before the break, seeing a couple of shots blocked and creating a moment of genuine excitement when Luis Sinisterra fed Kluivert and he crossed low for the run of Ryan Christie. Frustratingly for Bournemouth, he could not get a touch at point-blank range.

Klopp had to make do without two of his leaders – the injured Trent Alexander-Arnold and Mohamed Salah, who is on Africa Cup of Nations duty; he has picked up a hamstring problem to boot. The Liverpool manager, who missed plenty of others and lost Curtis Jones to a muscular problem towards the end, had started with Núñez on the left and Jota through the middle – the setup that had inspired the FA Cup win at Arsenal two weeks ago.

Klopp had Núñez in the centre at the start of the second half, Jota to the right, and it was a decision that paid off immediately. The move for the breakthrough goal was underscored by precision and technique, Jones firing it with a touch on his chest to control an Ibrahima Konaté crossfield ball and a smooth release to Jota on the bounce. He knew where Núñez was and he found him first time. Núñez knew where the bottom corner was and he found it first time. Liverpool turned the screw. Jota felt a touch from Zabarnyi on the very edge of the area and went down wanting a penalty, which did not come – it might have done – while Conor Bradley fluffed a header when well placed after Núñez had surged to the byline and crossed. Bradley emerged with honours at right-back and he would be credited with the assist for Jota’s second goal after crossing low.

The needle endured. Lewis Cook was booked for a hack at Bradley and a bit of afters during the untangling operation, which had Klopp exchanging angry words with the Bournemouth bench. Klopp could be happier with his team’s second-half performance as they pulled clear, bristling as it did with control and authority. He could also introduce a few of his youngsters off the bench – Owen Beck, Bobby Clark and Kaide Gordon.

Bournemouth had been promising in the first half but it would turn into an ordeal for them, their woes complete when the substitutes David Brooks and Kieffer Moore wasted glorious chances during the eight minutes of stoppage time.