Spartak had only conceded one goal in their opening two matches under Murat Yakin, Krasnodar were rampant at Stadion Kuban and collected their first win of the campaign in style.
Marat Izmailov got the scoring underway in the 27th minute with his first for the club, before three goals in 11 second-half minutes secured the points and lifted Krasnodar into the top six.
Early leaders Zenit maintained their 100 per cent record thanks to Salomon Rondon's 74th-minute winner at Ural, earning Andre Villas-Boas' side a 2-1 victory on Wednesday.
The only other side yet to drop a point this term are CSKA Moscow, with their latest victory coming at home to Terek Grozny as Zoran Tosic's first goal of the season sealed the points.
Kevin Kuranyi continued his impressive start to 2014-15 with his third goal in as many games as Dinamo Moscow condemned Ufa to defeat in their first ever top-flight home match.
Defender Christopher Samba opened the scoring on 32 minutes on Wednesday before Kuranyi doubled their lead a minute after the hour, with both goals beign provided by new signing Mathieu Valbuena.
Two of Dinamo's city rivals also picked up points as Alan Kasaev and Aleksandr Samedov gave Lokomotiv Moscow a 2-1 win over Rostov, while Torpedo Moscow got their first point of the season with a 1-1 draw at home to Amkar Perm.
Nikolay Savichev's Torpedo had conceded 12 goals in their opening two games, including eight against Zenit, but looked much more secure at the back on this occasion.
Meanwhile, Arsenal Tula also got off the mark with a goalless draw against Rubin Kazan and Mordovia Saransk were unable to make use of their numerical advantage as they drew 0-0 with 10-man Kuban Krasnodar.
