Liverpool will head into the international break with 28 points from 11 games with reigning champions City on 23.
The thrilling draw leaves Liverpool second, a point behind table-toppers Manchester City.
The 37-year-old shot hit the post as he sank to his knees in frustration.
In the second half, Liverpool improved, as Jürgen Klopp moved Núñez to the left side.
Haaland watched the encounter from the stand after being sidelined by an injury prior to the last international break.
With two minutes from the half-hour mark, Nigerian-born Philip Anyanwu Billing put Bournemouth ahead thanks to a pass from Ouattara’s brilliant counterplay, which eventually became the match-winning
Liverpool returned for the second half much stronger, doing more damage as the Red Devils completely lost control of the game.
Arsenal have gone five points clear off the top of the Premier League table following an emphatic 4-0 win over Everton at the Emirates Stadium.
Real Madrid demolished Liverpool on Tuesday night, leaving Anfield shell-shocked.
With the win at St James’ Park, Liverpool ended Newcastle’s 17-match unbeaten run.
