It was the home side that started the better and they had a chance to take the lead after three minutes when Kim Kallstrom whipped a dangerous cross into the box, but Mikael Lustig cleared the danger.
Celtic opened the scoring, however, when Gary Hooper finished from close range after 13 minutes. The Scottish side enjoyed a period of possession following the strike, but Spartak remained the side in the ascendancy.
Hooper had the ball in the net for the second time with 25 minutes on the clock, but the flag was raised following the striker's header.
Spartak's best chance of the opening half fell to Emmanuel Emenike after 35 minutes, but the striker could only head wide from close range.
Emenike had one last chance before the half, however, and the Nigerian found the back of the net from close range to level the scores.