The match, won 1-0 by Tula, was also marred by violence in the stands with Spartak fans climbing on to the roof of the stadium and throwing flares and other objects at rival supporters.
"Due to the public display by their supporters of a banner, which included racist symbols (a Celtic cross), Spartak have been fined 200,000 roubles," the Russian Football Union's website stated.
"The club will also be banned from having supporters attend their next two away games (this will not include women and children up to the age of 12) in the Russian Premier League."
Speaking to reporters, the head of the Disciplinary Committee, Artur Grigoryants, said: "Russia has signed up to all the necessary documents supporting the fight against racism and discrimination.
"A swastika and a Celtic cross are two of the main symbols, which are absolutely necessary to banish from the walls within a stadium along with banners with these emblems," he added.
Arsenal Tula were also fined 840,000 roubles with the bulk of that sum for failing to provide adequate safety measures, leading to a supporter falling off the roof of the stadium and breaking both his arms.
Arsenal will also be forced to play their next Russian Premier League home match at a neutral venue.
Spartak are currently in seventh place on 36 points, while Arsenal are in 13th on 20 points.
(Reporting By Dmitriy Rogovitskiy; editing Martyn Herman)
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