Zenit began life without Luciano Spalletti in a 1-0 defeat at CSKA Moscow while Anzhi's unlikely bid for survival gathered pace.
Spalletti – who led Zenit to two league titles after joining the club in December 2009 – was sacked last week after a goalless draw against strugglers Tom Tomsk.
That saw Zenit drop to second in the table, with Spalletti's departure coming after a run of one win in six league matches.
The club were unable to improve their form as interim head coach Sergei Semak took charge on Saturday.
Former Chelsea and Tottenham coach Andre Villas-Boas has been tipped to take over at Zenit soon, but that may come too late to secure another title if their poor form continues.
CSKA Moscow, who sit fifth, won on Saturday thanks to a first-half goal from Georgi Milanov.
Milanov's goal was his first for CSKA since joining the club last year from Lovech in Bulgaria.
Zenit remain second and are three points behind leaders Lokomotiv Moscow, who drew 0-0 with Amkar Perm.
A week after recording their first league win of the season, Anzhi came from two goals behind to draw 2-2 at third-placed Spartak Moscow on Monday.
Spartak seemed set for victory after Alberto Costa and Jurado scored in the first half.
But Anzhi – who were very unlucky to lose 1-0 at AZ in the first leg of their UEFA Europa League last-16 tie on Thursday, a result that ended an unbeaten run of three – showed great fight to win a point.
Diniyar Bilyaletdinov set up Alexandru Epureanu for a 52nd-minute goal and substitute Serder Serderov grabbed an equaliser in injury-time.
Serderov's 95th-minute goal means Anzhi are now four points behind second-bottom Terek Grozny.
Kevin Kuranyi scored for the fourth league match in a row for Dinamo Moscow, who had to settle for a 1-1 draw at Kuban Krasnodar on Saturday.
Kuranyi's 43rd-minute effort was cancelled out by Ivelin Popov with 17 minutes to play.
Armenia international Edgar Manucharyan scored twice in the last five minutes to see struggling Ural to a 2-1 victory over Terek Grozny on Friday.
Grozny were forced to play the majority of the game with 10 men following Fedor Kudryashov's 11th-minute dismissal.
Elsewhere, Krasnodar scored late to salvage a point in a 1-1 draw at Tom Tomsk and Rubin Kazan beat Volga 3-1, while Rostov defeated Krylya Sovetov 2-0.