Bulgarian striker Georgi Peev gave seventh-placed Amkar a shock first-minute lead when he turned in a Martin Jakubko cross from the right to delight a small band of away fans in the Khimki Arena.
Ivorian Doumbia levelled midway through the first half, superbly controlling a long ball from defender Vasili Berezutski before he beat keeper Roman Gerus with a cool finish from 12 metres.
Doumbia limped off with what appeared to be a thigh injury several minutes later and the loss seemed to rattle the hosts.
Konstantin Vassiljev nearly restored Amkar's lead when he hit the crossbar with a fierce long-range shot in the 44th minute.
With CSKA's bare-chested fans roaring on their team in the pouring rain, the champions laid siege to Amkar's goal in the second half and their efforts were rewarded when Nigerian Musa sent the home faithful into raptures.
The visitors failed to clear the ball after Gerus could only parry a stinging Zoran Tosic shot in the 77th minute and Musa popped up on the edge of the penalty box to curl a delightful shot into the far corner.
The result left CSKA on 17 points from seven games, with city rivals Lokomotiv and surprise package Rostov on 13 each ahead of their Sunday visits to Zenit St. Petersburg and Dynamo Moscow respectively.
CSKA are at home to Rostov in their next match on September 15 after the international break while Amkar host former champions Rubin Kazan.