The Ivory Coast striker opened the scoring with a low shot following good work by Brazilian forward Vagner Love in the 13th minute but Love's compatriot Danilo Neco equalised for Alania, playing in their first cup final, 10 minutes later.
It was Alania's first goal from open play in this cup campaign as three of their wins, including the semi-final against Rostov, came courtesy of penalty shootouts.
Doumbia headed the winner from close range after a pinpoint cross by Japan midfielder Keisuke Honda midway through the second half to seal CSKA's sixth cup triumph in the past decade.
"The cup final is never easy, no matter who your opponent is," said Slutsky, 40. "Obviously, I'm very very happy now."
As cup finalists, second division leaders Alania had already secured a place in next season's Europa League because CSKA have qualified for the Champions League.
The defeat was also bitter for former CSKA's long-serving coach Valery Gazzayev, who guided them to victory in the 2005 UEFA Cup among others before being sacked in 2008.
"I have nothing but praise for my team," said Gazzayev, at 30 the youngest coach in Russian professional soccer.
"CSKA have had a lot more experience playing in many such games and it made a big difference in the end."
