The Russian FA fined visiting Spartak 600,000 roubles (£11,500) and told them to play two home matches behind closed doors for their part in the scenes of disorder.
The FA also said fans of the club were guilty of displaying Nazi symbols.
Spartak, who are third in the top flight, will have to play second-placed Lokomotiv Moscow on Sunday and leaders Zenit St Petersburg seven days later in an empty stadium.
Second-tier club Shinnik were fined 500,000 roubles and ordered to play three home games behind closed doors.
Wednesday's tie in the city of Yaroslavl saw fans clashing with police, ripping out seats and hurling them on to the pitch along with flares and smoke bombs.
The game, won 1-0 by Spartak, was interrupted and police had to use a water cannon to restore order. A total of 78 supporters were detained by police after the match.
Earlier this week CSKA Moscow were ordered to close a section of their stadium for their next Champions League home game following the racist abuse hurled at Manchester City's Yaya Toure last month.