Come the end of the season Arne Slot will be hoping he can laugh about last week’s defeat to Nottingham Forest. Since that pitiful performance, Liverpool have made light work of Milan in the Champions League and gone top of the Premier League by pummelling Bournemouth in a perfect response to the first setback of the Dutchman’s tenure.

Related: Tottenham 3-1 Brentford, Liverpool 3-0 Bournemouth and more: football clockwatch – live

Where Forest made Liverpool struggle with a disciplined display, Bournemouth allowed Ryan Gravenberch and his midfield colleagues to dictate the match and then offered up an opener that ended the contest.

Luis Díaz took personal responsibility for causing a lot of the damage with two goals and Darwin Núñez, with a point to prove, added a third to banish the memories of seven days ago.

“After the big win against Milan, I was curious to see how we would react today,” Slot said. “And there was a much, much better reaction than we had against Nottingham Forest.”

Liverpool will have thought they were in for another testing match after an eventful opening four minutes, in which Ryan Christie was booked for a foul on Díaz withing 20 seconds of kick-off and Antoine Semenyo had a goal disallowed. The assistant referee saw no issue with the goal, only for VAR to silence those supporters who had made the 524-mile round journey from the south coast by acknowledging the clear offside at the back post after fine work by Justin Kluivert. The most galling aspect for Bournemouth was the lack of necessity for Semenyo to be ahead of play. Liverpool made the most of the reprieve.

• Download the Guardian app from the iOS App Store on iPhone or the Google Play store on Android by searching for 'The Guardian'.

• If you already have the Guardian app, make sure you’re on the most recent version.

• In the Guardian app, tap the Menu button at the bottom right, then go to Settings (the gear icon), then Notifications.

• Turn on sport notifications.

Kepa Arrizabalaga arrived at Bournemouth on loan from Chelsea in the transfer window desperately hoping to offer some evidence of why he is the most expensive goalkeeper in football history. He made a number of aesthetically pleasing saves to keep out two Díaz shots and was quick to smother a Mohamed Salah chance. All that was forgotten when Ibrahima Konaté lumped a long ball forward; Arrizabalaga thought he could come to claim it outside the box, only for him to come up short as Díaz rounded him and shot into an empty net. Prior to the goal Díaz was Liverpool’s most dangerous player with his usual mix of brute force and artistry, earning the luck for his reward.

The mistake was still playing over in Arrizabalaga’s head when Salah and Trent Alexander-Arnold combined on the halfway line. The full-back sprinted 30 yards up the pitch and slid a pass to Díaz who put the ball under the Bournemouth goalkeeper for Liverpool’s second goal in less than three minutes. If anyone was worried about a repeat of the defeat to Forest, their fears had been allayed.

All of Bournemouth’s problems were born in South America as Núñez followed Díaz’s lead with a one-two with Salah before heading down the right wing and cutting in, allowing him to bend a shot into the far corner for a third Liverpool goal in 12 minutes. The Uruguayan was clearly emotional as he celebrated in the corner in what was his first start of the season and first goal in 15 games.

Slot had backed Núñez to come good in the week and he repaid the faith. “My first instinct was when he shot, I said ‘why does he shoot? Why don’t you keep on dribbling?’” Slot said. “I saw the ball went in off the post. I was like ‘OK, maybe you are a better football player than I was in the past.’”

Bournemouth had as many shots as the hosts, although seven fewer on target, as they did what they could to get back into the game. Liverpool were, however, out of sight, allowing Slot to send on Federico Chiesa for a Premier League debut in front of a buoyant Anfield crowd and get Díaz and Núñez the standing ovations they merited, while Caoimhín Kelleher, replacing the injured Alisson, had a comfortable afternoon. Liverpool had their glitch but are back up and running, having won four out of five league games, conceding once.

For Bournemouth, however, they have won once in eight league matches. “The stats were quite even but I think Liverpool were better when it mattered,” the Bournemouth head coach, Andoni Iraola, said. “This is what big teams can do, when things were on the line, they finished better. When they had the chances, they made the difference with the finishes.”