Forgotten man Federico Chiesa, Liverpool's solitary signing last summer who has barely featured and whose future looked to be elsewhere, volleyed home his first league goal in the 88th minute before Mo Salah scored for the eighth time in nine opening-day fixtures.
Salah's strike meant the prolific Egyptian also became the joint fourth-highest scorer in Premier League history.
The hosts squandered a two-goal with new-signing Hugo Ekitike and Cody Gakpo scoring earlier.
Ekitike starting to pay back his £69million transfer fee with a first-half goal, having also scored last weekend in the Community Shield penalty shootout loss to Crystal Palace.
Within four minutes of the second half, Gakpo, via an Ekitike assist, doubled the lead to give the Reds some comfort as they struggled in patches.
In between those goals though the game was briefly paused after Bournemouth forward Antoine Semenyo reported to referee Anthony Taylor in the 28th minute he had been targeted with racist language by a member of the crowd.
Semenyo, who is Black, was consoled by players on both teams after the incident but played on and scored in the 64th and 76th minutes to draw Bournemouth level.
Liverpool manager Arne Slot was also seen talking to Semenya after the match.
An anti-discrimination message was read out to the crowd inside Anfield after the halftime whistle, the BBC reported.
The Football Association said it was "very concerned about the allegation of discrimination from an area of the crowd."
"Incidents of this nature have no place in our game," the FA added, "and we will work closely with the match officials, the clubs and the relevant authorities to establish the facts and ensure the appropriate action is taken."
It was the first competitive match at Anfield since Jota - a popular player for Liverpool - and his brother Andre Silva were killed in a car crash in Spain on July 3.
Ahead of kickoff, fans held up placards to spell out "DJ20" and "AS30" in two of the stands during a period of silence in honour of the Portuguese players.
Players from the Liverpool team stood arm-in-arm around the centre circle, and staff and players from both clubs wore black armbands.
Salah appeared to be tearful after fulltime as Jota continued to be commemorated by Liverpool's fans.
with PA
