Gordon Strachan brought in Scott McDonald to partner Jan Vennegoor of Hesselink up front, with Kenny Miller dropping to the bench after starting in the 4-1 weekend win at Falkirk.
The Celtic rearguard held firm in the opening minutes, as the SPL champions adjusted well to the synthetic playing surface inside the Luzhniki Stadium.
Twenty-one minutes in the Bhoys took the lead as McDonald found space wide on the right and delivered a quality cross into the centre of the penalty area.
Vennegoor of Hesselink drew his marker under the ball and Paul Hartley arrived right on cue to plant a superb 12-yard header beyond the despairing dive of Stipe Pletikosa.
Four minutes before the interval Spartak drew level as Roman Shishkin got away from his marker on the right and delivered an excellent left-foot cross to the back post, which was headed home in style by Roman Pavluychenko.
After the restart the home side continued to hold the whip hand, with the Bhoys forced to defend in depth at times.
Shortly before the hour mark, a corner caused havoc in the Celtic area, before keeper Mark Brown pounced on the loose ball.
Midway through the second half, goalscorer Pavluychenko headed into the ground and over after Welliton's cross had found him with time and space in the area.
Seven minutes from time Brown saved his side with a superb close-range block from Welliton's header, before Scott Brown spurned a clear chance at the other end.