New signing Hugo Ekitike marked his first league start with a goal in the 37th minute, before Cody Gakpo doubled the lead four minutes into the second half.
Bournemouth hit back through Antoine Semenyo, who scored twice in the 64th and 76th minutes to bring the visitors level. But substitute Federico Chiesa struck in the 89th minute to restore Liverpool’s lead, and Mohamed Salah sealed the win with a goal in stoppage time.
The match was briefly paused in the first half after Semenyo reported being racially abused by someone in the crowd. Bournemouth manager Andoni Iraola confirmed the incident, saying the person responsible had been identified. He described it as “a big shame” and praised Semenyo for reporting it immediately to the referee.
Liverpool manager Arne Slot was seen consoling Semenyo after the match.
The Premier League said the incident will be fully investigated, stressing that racism has no place in football or society. Both clubs have expressed their support for the player.