Inter, stunned by Turkey's Trabzonspor 1-0 at home in their Group B opener two weeks ago, triumphed thanks to Mauro Zarate's winner as Ranieri recorded his second victory in two games since replacing the sacked Gian Piero Gasperini.
"I think we beat a very good team tonight," Ranieri told a news conference.
"We played well, passed the ball well and also controlled the tempo," added the former AS Roma, Juventus and Chelsea manager.
"But the most important thing that led us to our victory was our team spirit."
Inter sold top striker Samuel Eto'o to Russian club Anzhi Makhachkala and signed Diego Forlan from Atletico Madrid in the close season but the Uruguayan is ineligible for the Champions League after playing for the Spanish side in the Europe League qualifiers.
Brazil defender Lucio's side-footed effort was deflected in following a corner after he was left unmarked at the far post and CSKA keeper Vladimir Gabulov misjudged a high ball.
Giampaolo Pazzini then added a second from close-range in the 23rd minute following a low cross by Yuto Nagatomo with CSKA's defence in disarray.
Ivory Coast striker Doumbia, who scored both of CSKA's goals in a 2-2 draw at Lille in their opener, also headed against the post midway through the fast-paced first half.
"I think we were a better team but luck was just against us," an emotional Dzagoyev told Russian television NTV Plus.
"I feel we deserved at least a point ton
