The lone point was enough to give the army side their first league crown since 2006 and 11th in the club's history as they now lead last season's champions Zenit St Petersburg by six points.
Zenit, who have two games left and face Volga Nizhny Novgorod on Sunday, could not overtake CSKA because they have a fewer number of total victories.
CSKA coach Leonid Slutsky paid tribute to his team's fans.
"This victory is for you because you have supported us the whole season," he told thousands of jubilant supporters, who filled the Khimki arena in suburban Moscow.
On a hot afternoon the home side took the initiative from the start. CSKA missed several chances in the first half, Nigerian Ahmed Musa and his Brazilian strike partner Vagner Love were the biggest culprits.
Musa wasted another opportunity midway through the second half when he was left unmarked in front of the goal after good work by Russia international Alan Dzagoyev, but his shot was blocked by Kuban defenders.
With time running out both teams looked to be content to settle for a draw. CSKA keeper Igor Akinfeyev stopped a close-range header from Kuban's Senegalese striker Ibrahima Balde in the 89th minute to preserve the vital point for his team and start the celebrations.
"This is my first title so it's hard to describe how I feel," Dzagoyev told Russian television NTV Plus.
"I can't describe it. Just happiness. The only problem was that we couldn't score, although we came close several times."
It was the first major title for Slutsky, who replaced Spaniard Juande Ramos near the end of the 2009 season.
Slutsky will now have a chance to lead CSKA to their first league and cup double in seven years when they take on Guus Hiddink's Anzhi Makhachkala in the Russian Cup final in the Chechen capital Grozny on June 1.