Kvyat will return to the Red Bull-owned Toro Rosso team that he started out with in 2014 while 18-year-old Verstappen moves up to the former world champions alongside Australian Daniel Ricciardo.
The next race, fifth of the 21 round championship, is the Spanish Grand Prix in Barcelona on May 15.
"Max has proven to be an outstanding young talent. His performance at Toro Rosso has been impressive so far and we are pleased to give him the opportunity to drive for Red Bull Racing," said team principal Christian Horner.
"We are in the unique position to have all four drivers across Red Bull Racing and Toro Rosso under long term contracts with Red Bull, so we have the flexibility to move them between the two teams," he added.
Horner said Kvyat, 22, could "continue his development... in a team that he is familiar with, giving him the chance to regain his form and show his potential."
Toro Rosso principal Franz Tost said they were "more than happy" to have Kvyat back.
'AMAZING OPPORTUNITY'
Verstappen was 2015 Rookie of the Year and is considered a champion of the future. Red Bull, determined to stave off interest from rivals, had been expected to promote him anyway at the end of the season.
Verstappen hailed his promotion as "an amazing opportunity", but Kvyat's demotion still came as shock even in a sport not known for second chances.
"Really? one bad race and Kyvat's dropped, what about the podium in the previous race? #Shortmemories," commented McLaren's Jenson Button on Twitter.
The only Russian currently on the starting grid, Kvyat has plenty of admirers but has also become embroiled in controversy.
He has clashed twice in two races with Ferrari's Sebastian Vettel, the four times world champion whose place he took at Red Bull, and has struggled to match Ricciardo's pace.
Vettel branded him a 'madman' in China last month for an overtake at the start that triggered a collision between the Ferraris.
Kvyat was named Driver of the Day after finishing third in that race, but he had no defenders after colliding twice with Vettel in Sochi last weekend in an accident that also ruined Ricciardo's race.
"The bottom line is he misjudged it, he hit the back of Seb, that concertina-ed into Daniel and that was pretty much the end of our afternoon," Horner said then.
(Editing by John O'Brien)
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