Salah scored a classy hat-trick against Eddie Howe's side at the Vitality Stadium as he looked back to his best form.
Salah was awarded the man-of-the-match award for his role in the Reds' victory, however he refused to accept it, instead handing it to Milner for his 'amazing' achievement in the game.
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"No, will not accept it," Salah said as he was handed the man-of-the-match award.
"I have to congratulate him for his amazing career. He deserves it today.
"He's had an amazing career and I hope we win something together this season.
"But no, I will not take it."
It was a classy gesture from Salah, with not many players in the world deciding to reject the award after scoring a hat-trick.
Milner looked equally embarrassed and grateful to his teammate and it showed yet another example of the solidarity the Reds squad have between themselves.
Salah played down his role in the victory, insisting what is most important is winning each game as it comes.
"My expectation is very high, everyone else has expectation very high," Salah said.
"The most important is we are top of the table and win each game. That's the most important thing."
The Reds are now on 42 points after 16 games, their best ever start to a league campaign as they look to chase down leaders Manchester City at the top of the table.