Two second half goals from the Liverpool striker have all but secured three points.
Milan Baros missed a couple of early chances to put the Reds ahead before Alexander Danishevsky gave the home side the lead on 23 minutes.
Michael Owen equalised with a header 6 minutes later as Liverpool started to get back into the game, and an impressive second half display earned them the victory.
Spartak: Levitsky; Kovtun, Beschastnykh, Kalinichenko, Danishevski, Besrodny, Pavlenko, Mitreski, Sonin, Abramidze, Khlestov.
Subs: Cherchessov, Kebe, Tchuisse, Da Silva, Kudriashov, Torbinsky, Essien.
Liverpool: Dudek; Traore, Hyypia, Carragher, Riise, Murphy, Hamann, Diao, Baros, Heskey, Owen.
Subs: Arphexad, Babbel, Smicer, Diouf, Biscan, Vignal, Mellor.
Referee: Konrad Plautz (Austria)