Saturn sub Andrei Movsesyan scored a dramatic winner with a diving header in the 90th minute after a cross from Moldova international Radu Rebeja.
Former Russia fullback Igor Yanovsky had given CSKA the lead after 20 minutes but Moldova international striker Sergei Rogachev equalized less than a minute later.
CSKA captain Sergei Semak again put his side ahead after an hour only for Pyotr Bystrov to level.
The result ended CSKA's six-game unbeaten streak as last year's runner-up Lokomotiv Moscow moved to the top with a 3-0 win over city rival Dynamo.
It was the first defeat for newly appointed Dynamo coach Viktor Prokopenko as his team was brought down to earth after thrashing lowly Rostselmash Rostov 5-0 six days ago.
Nigerian striker James Obiorah opened the scoring after eight minutes and Ruslan Pimenov added two more either side of halftime.
Pimenov's second, a right-foot volley into the top corner, strengthened his claim for a recall to the Russian team.
Loko goalie Sergei Ovchinnikov, out of the Russia World Cup squad following a spat with team coach Oleg Romantsev, had another strong game with his sixth clean sheet in seven matches.
On Friday, a lucky bounce helped champion Spartak Moscow to a 2-1 win over cross-town rival Torpedo.
A shot by Ukrainian midfielder Eduard Tsykhmeistruk from the edge of the penalty box hit a defender and bounced over sprawling Ukraine international goalkeeper Valery Vorobyov to give third-placed Spartak the points.
But the win proved costly as Spartak's Russian internationals Yury Kovtun and 18-year-old striker Dmitry Sychyov were forced off with injuries.
A Message from The Moscow Times:
Dear readers,
We are facing unprecedented challenges. Russia's Prosecutor General's Office has designated The Moscow Times as an "undesirable" organization, criminalizing our work and putting our staff at risk of prosecution. This follows our earlier unjust labeling as a "foreign agent."
These actions are direct attempts to silence independent journalism in Russia. The authorities claim our work "discredits the decisions of the Russian leadership." We see things differently: we strive to provide accurate, unbiased reporting on Russia.
We, the journalists of The Moscow Times, refuse to be silenced. But to continue our work, we need your help.
Your support, no matter how small, makes a world of difference. If you can, please support us monthly starting from just $2. It's quick to set up, and every contribution makes a significant impact.
By supporting The Moscow Times, you're defending open, independent journalism in the face of repression. Thank you for standing with us.
Remind me later.