Spartak, down a number of first-team players, surprisingly opened the scoring after 23 minutes, with Brazilian midfielder Fernando bending a free kick past Loris Karius.
It didn't take Liverpool long to respond however, as Philippe Coutinho played a delightful one-two, albeit against a static defence, that lead him into the box to fire past Artyom Rebrov on the 31st minute mark.
The Reds then dominated the second half, but their poor finishing and final ball cost them dearly.
Daniel Sturridge, who appeared rusty during his twenty minutes on the pitch, was unnmarked in the box, but he skied his volley from close range.
Mohamad Salah had a golden opportunity to steal the game at the death, but his header from inside the six-yard box was saved by sub goalkeeper Aleksandr Selikhov.
Jurgen Klopp was left frustrated with the result and another game having too come from a goal down.
However, the number of chances his side created would have pleased the German as they look forward to hosting Maribor at Anfield in their next Champions League fixture.