It is Liverpool's second 7-0 win in the group following their victory by the same scoreline against Maribor in Slovenia in October.
A deflected shot soon after saw Coutinho complete his hat-trick, with Mane scoring his second and the Reds' sixth before Salah completed the scoring to condemn Spartak to a demoralising loss on Merseyside.
He sneered: "If Liverpool were playing in my back garden, I wouldn't watch them".
"Liverpool won't win the Champions League because they don't have a defence".
Coutinho scored his first hat trick for Liverpool and it started when he stroked home a penalty after Georgi Dzhikiya tugged Salah in the penalty area.
Ironically, the Reds found themselves up 3-0 at halftime against their Russian opponents; the same scoreline as their last European fixture against Sevilla before the La Liga side roared back to take a share of the spoils with three goals of their own in the second half.
Milner, on as a substitute, crossed from the left for Mane to meet with a fierce volley past goalkeeper Aleksandr Selikhov in the 47th. We will see where this leads to, but it's a fact that English football is top quality.
Liverpool have made a formal complaint to Uefa after striker Rhian Brewster was the victim of alleged racial abuse by a Spartak Moscow player during today's game between the clubs.
Liverpool's prize for winning their group, and reaching the knockout stage for the first time since 2008-09, will be a last-16 tie against either Basel, Bayern Munich, Juventus, Porto, Shakhtar Donetsk or the reigning champions Real Madrid.
Spartak finished third in the group and dropped into the Europa League.
Coutinho's summer was dominated by his desire to move to the La Liga giants, so much so that he tried to force it with a late transfer request that was ultimately denied by Liverpool.