Doumbia twice capitalised on woeful defending to score, the first inside two minutes, either side of a sweet strike by Yaya Toure who was later sent off.
Halftime substitute Fernandinho was sent off midway through the second half before Toure followed after lashing out at a CSKA player to complete a dreadful night for City who have taken two points from four Group E games.
Russia's CSKA, who were banned from selling tickets for the game following repeated bad behaviour from their supporters, have four points, level with AS Roma and eight behind Bayern Munich who beat the Italian side 2-0.
"It is very difficult to understand how we played so bad from the beginning. We were so nervous, we gave away two bad goals. We didn't play," City manager Manuel Pellegrini told Sky Sports.
"It is difficult to know why it was such a low performance from important players. It is a strange moment. It is not just this competition."
Despite not having won a game in this year's competition, Pellegrini had told his pre-match press conference that his expensively-assembled side had targeted winning the Champions League.
But his ambitions were dealt an early blow.
Less than two minutes had elapsed when Doumbia, who scored CSKA's first goal and won the penalty for the equaliser in last month's 2-2 draw in Moscow, took advantage of lackadaisical defending to crunch a header past a static Joe Hart.
Toure was at fault for allowing Doumbia space for the opening goal but he more than redeemed himself curling a beautiful free kick into the top corner six minutes later.
With City struggling to play their slick flowing football CSKA fashioned a number of chances on the break and after Doumbia had blazed wide, he added his second.
The Ivorian had scored twice in CSKA's visit to the Etihad Stadium in last season's competition and added another brace, capitalising on more woeful defending to finish neatly past Hart.
After the break Diminutive City striker Sergio Aguero, who had scored five goals in his previous three appearances against CSKA, was agonisingly close to converting James Milner's teasing cross.
With City dominating possession but struggling to fashion any clearcut opportunities they were reduced to 10 men when Fernandinho received two yellow cards, the first for a wild lunge and the second for obstruction.
With frustration rising Toure lashed out and was shown a straight red card before Aguero was booked for diving in the shadows of fulltime to cap an awful evening for City.
