Individual 400 metres champion Natalya Nazarova brought them home in three minutes 23.88 seconds, beating the previous mark of 3:24.25 set on March 7, 1999, at the world championships in Maebashi, Japan.
The quartet also included Olesya Krasnomovets, the individual 400 metres silver medallist at these championships.
Belarus came second and Romania third.
It was the third world record at the indoors event, all of them set by Russia's women squad, following new marks in the women's pole vault and triple jump on Saturday.
Shortly before the relay, Sweden's Christian Olsson had equalled the men's triple jump world record of 17.83 metres to win the event.
