Fans launched a fire cracker on to the pitch in Slovakia and the teams were briefly led off by Dutch referee Kevin Blom after three minutes.
There were no apparent injuries after the missile came from an area of the stadium populated by visiting supporters.
During the suspension of play Spartak went to the end of the ground occupied by their fans and seemed to remonstrate with them.
The trouble came less than a week after Russia were elected as 2018 World Cup hosts following a vote by FIFA's executive committee. Their critics had said hooliganism and racism were still prevalent within the Russian game.
Local news agency SITA reported that some 400 Spartak supporters attended Wednesday's match.
Tomas Majtan put Zilina ahead in the 48th minute but Alex and Ibson replied for Spartak.
The Moscow team finished third in the group with nine points and qualified for the Europa League. Zilina were bottom with zero points.
Spartak dominated most of the match but fell behind against the run of play when Majtan struck.
Alex equalised in the 54th minute, pouncing on a superb pass with the outside of his right boot from substitute Nikola Drincic.
The lively Ibson made it 2-1 from close range in the 61st minute but blotted his copybook with 13 minutes to go when he was sent off.
