A goal by the right back Kirill Nabakin retained CSKA's perfect start to the campaign, leaving them four points clear of the second-placed city rivals Lokomotiv, who beat FC Krasnodar 2-1.
Reigning champions Zenit sit third two points further back.
After a goalless first half, CSKA got a chance to open the score in the 51st minute when they were awarded a penalty for a foul on Ahmed Musa.
But Israeli international Bebars Natcho failed to find the net from the spot, sending the ball over the crossbar.
CSKA upped the tempo, pinning Kuban back and their Serbian playmaker Zoran Tosic and full back Vasily Berezutsky missed opportunities as both hit the woodwork with powerful long-range shots.
In the 88th minute, Nabakin finally netted his side's winner from close range after a rapid counter-attack.
Kuban's Paraguayan defender Lorenzo Melgarejo had a chance to save the day for his side in injury time after Roman Bugaev was brought down in the box, but CSKA goalie Igor Akinfeyev parried his spot-kick.
"In the first half we failed to create any serious chances as we looked emotionally exhausted after Wednesday's Champions League clash with Sporting," CSKA manager Leonid Slutsky said.
"After the break we looked much better and dominated play.
"But after we scored, we were lucky at the very end of the match. Igor (Akinfeyev) was a true hero today."
Elsewhere, two-time former champions Lokomotiv Moscow battled back from a goal down to grab a hard-fought win to go second.
Russian international forward Fedor Smolov put Krasnodar 1-0 up in the 11th minute beating Lokomotiv keeper Marinato Guilherme with a low shot from the edge of the area.
Senegalese striker Baye Oumar Niasse pulled the scores level seven minutes later when he swept the ball home under goalie Andrei Dikan's hand.
Midfielder Alan Kasaev scored the railway-backed team's winner in the 56th minute when he headed home a rebound after Dikan parried Niasse's initial shot.
"It was a very difficult game," Lokomotiv head coach Igor Cherevchenko said.
"We were a bit shocked by Krasnodar's early goal. Luckily, we kept our nerve and soon levelled.
"After the break we upped the tempo and scored the winner. I'd say it's a deserved win."