Mane, Salah and Firmino combined for a superb one-touch move in the 15th minute, the latter squaring for Brazil team-mate Coutinho to finish back across a helpless Alexander Selikhov. The number nine spotted the run of his Brazilian counterpart at the edge of the box and delivered an inch flawless pass for Coutinho, whose left-footed shot sent Spartak goalkeeper Alexander Selikhov the wrong way.
Despite inexplicably throwing away a three-goal lead against Sevilla in the last round of the Champions League group stage, Liverpool still sit top of Group E with their fate firmly in their own hands.
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.
Liverpool fans are just enjoying him while they can. It also made Liverpool the highest scoring English team in a Champions League group campaign, the second highest ever, and Klopp admitted the attacking prowess of Coutinho, Mané, Salah and Firmino would intimidate every team in Europe providing it is maintained. We will see where this leads to, but it's a fact that English football is top quality.
Coutinho added another two minutes later, latching onto another Milner cross and firing another left-footed effort in via a Spartak defender to make the score an astounding 5-0 by the 50th minute.
Promes has been impressive this season with 13 goals and eight assists in 23 games.
Liverpool, which is in the knockout stage of the Champions League for the first time in nine years, could still be paired with Real Madrid, Juventus or Bayern Munich in Monday's draw for the last 16.
Spartak finished third in the group and dropped into the Europa League.