City manager Manuel Pellegrini had described the Group E clash as a must-win game, but a late Bebras Natcho penalty denied the visitors at an Arena Khimki without supporters due to UEFA sanctions following issues of racism and crowd trouble.
Goals from Sergio Aguero and James Milner had looked set to prove the difference for City in the first match of their double-header against the Russian side.
The English champions moved ahead after 29 minutes when the in-form Aguero continued his goalscoring streak with a simple finish, before Milner added a second seven minutes prior to the break.
With City on course for a first European win since last December, Seydou Doumbia pulled one back for Leonid Slutsky's side and, as the home side increased the pressure, they were rewarded when a sloppy challenge from Aleksandar Kolarov on Ahmed Musa resulted in a spot-kick being given with four minutes remaining.
Natcho converted to rescue his side's first point of the campaign and ensure City missed the chance to apply pressure to Bayern Munich and Roma.
Musa scored twice in CSKA's 6-0 win over Kuban Krasnodar at the weekend and almost opened the scoring on Tuesday with a fizzed effort that veered wide of Joe Hart's left-hand post.
Former Manchester United man Zoran Tosic was next to threaten 10 minutes in as City, having made five changes from Saturday's 4-1 win over Spurs, looked somewhat disjointed early on.
Pablo Zabaleta - among the players who returned to the starting XI - shot wide after expertly controlling Yaya Toure's lofted ball but, in an opening half-hour of few clear-cut chances, it was Aguero who again provided the breakthrough for City.
Silva slipped in Dzeko, whose square ball laid on Aguero's eighth goal in his last five outings.
The Argentina international then played a key role as City doubled their advantage, his wayward strike - following Zabaleta's knockdown - finding Milner at the far post, who turned home to double the visitors' lead.
Milner subsequently struck the post with the goal at his mercy, having seen Dzeko's appeals for a penalty turned down following a tangle with Sergei Ignashevich.
CSKA's day looked set to worsen when Doumbia appeared to pick up an injury, but the Ivorian then gave the hosts hope by turning home Musa's cross from the right five minutes after the hour.
City's failure to put the game to bed was then punished when Musa went down under a challenge from Kolarov, giving Natcho the opportunity to equalise with a nerveless penalty that beat Hart as the goalkeeper dived to his left.
CSKA were therefore able to get off the mark at the third attempt and inflict a heavy blow to their opponents' hopes of reaching the last 16.
