Spartak are level on 10 points with surprise package Rostov, who won 1-0 win at big-spending Anzhi Makhachkala on Friday to pile the pressure on the hosts' coach Rene Meulensteen.
Nigeria striker Emmanuel Emenike fired Spartak ahead against the run of play at Kuban, heading home an inch-perfect cross by Spanish midfielder Jose Manuel Jurado after goalkeeper Sergey Pesyakov denied the hosts the lead with several good saves.
Kuban turned the tide midway through the second half when Senegalese striker Ibrahima Balde raced clear to equalise and Brazilian defender Xandao gave them the lead as he drilled home a rebound after Pesyakov parried Gheorghe Bucur's header.
Substitute Yura Movsisyan brought Spartak level and helped maintain their unbeaten start with a 75th-minute thunderbolt from 25 metres which crashed in off the underside of the bar and left Kuban keeper Aleksandr Belenov rooted to the spot.
Chances were few and far between at Kazan, where Russia keeper Igor Akinfeev kept CSKA afloat with a string of superb stops as the champions were pegged back by a more adventurous Rubin side.
Salomon Rondon, Vladislav Kulik and Oleg Kuzmin all came close for the hosts while Swiss midfielder Steven Zuber missed CSKA's best chance when he rattled the woodwork with a screamer from outside the box.
CSKA are two points behind the leading pair while Zenit St Petersburg climbed to fourth on seven points after Portuguese striker Danny scored a hat-trick in Saturday's 3-1 victory at Volga Nizhny Novgorod.
Lokomotiv Moscow are at home to FC Krasnodar in the round's final game on Monday evening.