Trailing 3-1 with six minutes to go, the English champions set up an exciting finale when Paul Scholes headed in a Gary Neville free kick.
A frantic ending ensued and United salvaged a point with a last-gasp deflected strike from Antonio Valencia.
The draw put United through to the knockout rounds with two games to spare.
They have 10 points from four games, three more than second-placed VfL Wolfsburg who beat Besiktas 3-0. CSKA are third on four points, with Besiktas bottom with one.
"If you want excitement, come to Old Trafford! One of the great pleasures of being a manager here is when you get finishes like that," United manager Alex Ferguson told reporters.
Tomas Necid chested the ball to his strike partner who ran at goal and just as it looked as if United's Jonny Evans had pushed him too wide to shoot, Dzagoyev launched the ball into the roof of the net for a stunning opener.
POACHER'S FINISH
Michael Owen equalised four minutes later with a typical poacher's finish from close range.
"He hadn't had much sleep. I would have preferred to leave him on the bench," said Ferguson.
"He was a safety player that we knew we could have played if we needed him - he was there. I think his energy and confidence was very high. He makes an impact."
The Russian side, with coach Leonid Slutsky in his first European match in charge after last week's dismissal of Juande Ramos, could not hold on as Scholes beat Igor Akinfeyev with his point-blank header and Valencia grabbed the equaliser via a massive deflection.
