Zenit, managed by former Tottenham boss Andre Villas-Boas, hosts Monaco on Wednesday in Group C of the Champions League.
CSKA made hard work of beating Ural, only clinching the three points with Seydou Doumbia's 89th-minute winner. It could be matched on 21 points Sunday by either Dynamo Moscow or Kuban Krasnodar, which play each other.
CSKA and manager Leonid Slutsky have now won four of their last five matches in all competitions but are under pressure in the Champions League after a 5-1 thrashing by AS Roma in their Group E opener.
Tuesday's home game against Bayern Munich will be played in an empty stadium as UEFA's punishment for racist behavior by CSKA fans last season.
CSKA also faces further UEFA charges over alleged fan violence and racist incidents during the Roma game.
Also Saturday, FC Ufa rose to ninth with a 2-0 win over fellow newly promoted side Mordovia Saransk.