Minutes later, Coutinho completed his hat trick from close range courtesy of a deflection off of a Spartak defender that left Alexander Selikhov no chance in goal.
There are some very good teams lying in second place in their respective groups, who Liverpool might face in the next round, but this performance sent out a warning to all teams in Europe that they could struggle to get past them, even though they're a side which hasn't been in the Champions League knockout stages since 2009.
Chelsea, Manchester City, Manchester United and Tottenham will join the Reds in Monday's draw at Uefa headquarters in Nyon, Switzerland.
The former Twente star has a close relationship with worldwide colleague and Liverpool man Georginio Wijnaldum, and in January he admitted the Reds man wanted him to move to Anfield.
This time there was no relenting as Liverpool turned it on in the second half, their attack showing a level of understanding and incisiveness that will make potential rivals in the last 16 take notice.
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Rarely is such an attacking need required by the Reds but Klopp flexed his muscles in showcasing his four stars to the rest of the Champions League - and Premier League. Milner delivered an inviting cross for Mane to thump home on the volley less than two minutes into the second half.
But having given up a 3-0 half-time lead in their last Champions League outing in Sevilla, Liverpool were in no mood to drop down a gear and they put the game to bed within five minutes of the restart.
Three minutes later, Liverpool added their third.
When the Champions League resumes in February, it will be a first time since 2009 that Liverpool have reached the knockout round of the Champions League, with only bruising early exits to show from their two group campaigns across the intervening years.
Prior to this game there were still doubts as to whether Liverpool had what it takes to qualify.
Liverpool can secure a first place finish in their group with a win tonight.
A Technical Strategy for Bank of America Corporation (BAC)
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