The hosts were the first to threaten when Dmitri Kombarov shot over from inside the area after only ten minutes.
But the visitors broke the deadlock midway through the first half when Zhirkov jumped on a loose ball, smashing an unstoppable drive past the helpless Andriy Dykan.
Spartak despite plenty of possession could not carve a clear opportunity with Welliton cut a forlorn figure up front.
While Anelka was causing Spartak's back four all sorts of problems with his movement, and he could have doubled his side's lead on 38 minutes.
However the French striker made no mistake two minutes before half-time, as he slotted the ball past Dykan, having been found by a fabulous through ball from Michael Essien.
Spartak upped the tempo in the second half, with Aiden McGeady forcing Petr Cech into making a couple of smart saves. But for all their intricate interplay, the home side never really looked like reducing the deficit, and had to resort to long-range efforts.