After the original fixture was postponed due to heavy snow, the re-arranged game saw visitors Grozny take the lead through Jeremy Bokila, only for the hosts to hit back on the stroke of half-time via Luis Caballero's sublime finish.
The draw leaves the home side in 11th place, two points clear of the bottom four and a spot in the relegation play-offs with six rounds of action remaining.
Grozny occupy one of those berths, currently in 13th, just one point behind 12th-placed Ural and two behind their opponents on Tuesday as the race to avoid the drop looks set to go down to the wire.
After a quiet opening, Emin Makhmudov went close for the home side in the 34th minute, seeing his downward header held by Yaroslav Godzyur.
They were left to rue that miss six minutes later as Bokila got on the end of Ismail Aissati's right-wing cross to poke the ball in off the left-hand upright from six yards.
The lead lasted just four minutes, however, as Makhmudov teed up Caballero, who arrowed a shot in from 20 yards to send the sides in to the break level.
In a turgid second half, neither side managed to create any clear goalscoring opportunities as both were forced to settle for a point, Sovetov ending a run of four consecutive league defeats.
