From Mane's pass, Salah sent the ball out to Firmino on the right and his selfless square ball allowed Coutinho to tuck away his 50th goal in Liverpool colours.
Spartak's defending was as desperate as Liverpool's attacking was dazzling and a clumsy touch by Serdar Tasci opened the door to a third goal.
Unable to react quickly enough to Mane's cross, the German centre-back inadvertently presented the ball on a plate to Firmino, who used the outside of his right foot to steer a shot past Aleksandr Selikhov.
It was not all plain sailing for Klopp's men, however.
Can picked up a seventh-minute booking that triggered a one-game ban for sliding in on Ze Luis and Moreno hobbled off just before half-time after jarring his knee as he blocked a shot from Roman Zobnin.
James Milner took Moreno's place and made an immediate impact in the second half, swinging over a cross in the 47th minute that was brilliantly volleyed home by Mane at the back post.
Three minutes later Coutinho added a fifth, charging onto Milner's pass and dispatching a shot that found its way past Selikhov via a deflection off Salvatore Bocchetti.
Mane made it 6-0 with 14 minutes remaining, hooking in a low cross from substitute Daniel Sturridge, before Salah completed a resounding win with a sublime turn followed by a right-foot finish into the top-right corner.
Spartak coach Massimo Carrera delivered a damning assessment of his team's display.
"The players almost played like they were afraid," said the Italian, whose side dropped into the Europa League.
"It proved Spartak aren't ready for this level of football. Today we lacked quality in all aspects of football."
Everton, who visit Anfield in Sunday's derby, were left with plenty of food for thought.