Royals owner Zingarevich took control of the club from Madejski last year and became a close friend of McDermott's.
But a major decline in form ended with Zingarevich axing McDermott in March before appointing Nigel Adkins as his replacement.
Reading are yet to win in five games under Adkins and were relegated from the Premier League on Madejski's 72nd birthday on Sunday after a 0-0 home draw with QPR, who also went down.
But Madejski insists the Royals are in safe hands in four-time promotion winner Adkins.
"I think Anton has chosen really well with Nigel, I really do," declared Madejski. "I think he's a very enthusiastic, able person who can get us back into the Premier League.
"It's a different type of football that he plays and Anton didn't just make his mind up overnight.
"Now we can start building our team over the summer. We've got a new manager and we have to recognise that he's got his plans.
"He's been working with the team for a month or so. He knows how it all works and exactly what he's got to do."
He added: "Hopefully we can fight and get back into the Premier League next season. I'm delighted that Anton has bought in to the Reading way.
"He has great ambition for Reading Football Club, he wants us to do well.
"Now he's got his stall laid out, he knows what he wants, where we're going and where we've come from. I think it's a tremendous relationship we have with Anton."
The cost of relegation from the Premier League is estimated at around £60 million, with Reading now set to miss out on a chunk of the new £3 billion television deal which comes into force in the top flight next season.
It means Adkins will have to trim the wage bill and possibly sell players over the summer in a bid to reduce costs.
The blow will be softened by an increase in parachute payments from £48 million to £60 million over the next four years.
Royals will receive £24 million next year, £18 million the year after and £9 million in each of their final two years.
Zingarevich is also likely to pump in his own cash as he plans to increase his majority 51% stake in the Royals, which he bought for £25 million last year.
It will give Royals a fighting chance of winning promotion from the Championship next season, and Madejski stated: "The players certainly have the bit between their teeth to succeed next season and we as a club will be doing everything we can to get back into the Premier League.
"The whole club is focused on that and I think the fans also are too. We still have the good will of the fans which is vastly important for us.
"I think that the fans appreciate that Reading always have punched above their weight and I think that's important too.
"Sometimes it's difficult, when you've got 20 teams in the Premier League who are all desperate for the same accolade - to stay there. We've had this experience before, we know how to get back up again."