McDermott was relieved of his duties after a run of four straight Premier League defeats left the club off the bottom of the table on goals scored only.
Coaches Nigel Gibbs and Yannis Anastasiou have also departed, leaving Academy boss Eamonn Dolan in charge for this weekend's trip to Manchester United.
Dolan met the first-team squad on Tuesday along with Zingarevich, who completed his takeover of Reading just weeks after McDermott had masterminded promotion from the Championship.
And a statement on the Reading website read: "Anton told the players it was the hardest decision of his life, thanking all of the staff for their contribution.
"He and Brian remain on good terms after meeting in person on Monday.
"Eamonn held a series of meetings throughout the day in preparation for Manchester United, and he is scheduled to speak to the media later this week in the build-up to the game."
Paolo Di Canio, who left Swindon last month, is the current favourite with Sky Bet to replace McDermott at the Madejski.