The reigning champions, who have won three games in a row, are unbeaten in seven matches and now sit three points clear after Lokomotiv Moscow lost.
Amkar raced into the lead inside the opening minute through veteran Bulgarian Georgi Peev, but CSKA bounced back with a goal in each half.
Seydou Doumbia levelled before the interval and Ahmed Musa got the winner 13 minutes from time.
Spartak Moscow and Zenit St Petersburg made sure they kept in touch by picking up three points apiece on Sunday.
Zenit made Lokomotiv, who had occupied second spot, pay for going down to 10 men after ex-Real Madrid man Lassana Diarra was sent off in the first half.
Roman Shirokov scored from the resulting penalty but Lokomotiv's Mbark Boussoufa levelled from the spot for the away side after the break.
Midfielder Konstantin Zyryanov fired in a winner four minutes from time to move Zenit above the visitors.
In Saturday's games, strugglers Terek Grozny and Anzhi Makhachkala shared the points in a 1-1 draw, and Krasnodar thumped 10-man Volga 3-0, with Brazilian Wanderson scoring twice.
Rubin Kazan finally ended a sequence of four draws, in which they had only scored one goal, by beating Ural 3-0 away from home on Sunday.
Salomon Rondon netted a hat-trick to give Rubin fans something to celebrate but Ural will be concerned that they have not win since beating bottom-placed Tom' Tomsk at the beginning of August.
Tomsk, though, in their clash with Spartak Moscow went one up after seven minutes before Yura Movsisyan equalised and then Pavel Yakovlev got his third goal of the season to win the match.
Krasnodar's three-match winning run in all competitions came to an end after a 0-0 draw away to Krylya Sovetov.
In Sunday's final game, Rostov dropped a place after their 1-1 draw against Dinamo Moscow.
Hrvoje Milics's seventh-minute strike was cancelled out by Aleksandr Kokorin's second-half goal at the Arena Khimki.
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