It was his second hat trick for Liverpool since arriving from Roma last summer, but he felt Milner, who made his 500th Premier League outing, was more worthy of the accolade.
Liverpool, now enjoying their best start to a domestic season having accrued 42 points from their opening 16 games, moved top of the Premier League with the victory, one point ahead of Manchester City who face Chelsea later on Saturday.
Things got worse for Cook, who scored an own goal to make it 3-0 to Liverpool.
Liverpool struck just when the hosts were enjoying their best spell, though, as Asmir Begovic failed to hold on to Firmino's long-range shot and parried the ball out into a risky area, where Salah was waiting to tuck home the rebound.
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Both Bournemouth and Liverpool are coming off impressive wins midweek.
Klopp's side then doubled their advantage 3 minutes in to the second half as Salah picked up the ball in midfield and raced in to the opposition box.
"The problems he causes defenders, running on the shoulder, the third goal epitomised that".
In doing so, Salah drew level with Pierre-Emerick Aubameyang as the Premier League's top scorer this season, and he was given the chance to see out the game by Klopp despite matches against Napoli and Manchester United next week.
Jurgen Klopp's side went into the game just two points behind Manchester City, with City travelling to Chelsea on Saturday evening. Sadio Mane is a major doubt, but Georginio Wijnaldum and Andy Robertson is back in the reckoning.