Red Bull is yet to confirm if Verstappen will retain the seat for the remainder of the season.
The switch has come after Kvyat attracted heavy criticism at Sochi for causing avoidable contact with Ferrari's Sebastian Vettel, which led to the four-time world champion's retirement on the opening lap.
His error also inflicted damage to Red Bull team-mate Ricciardo having been collected by Vettel's out of control Ferrari.
“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 Red Bull boss 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.
“Dany will be able to continue his development at Toro Rosso, in a team that he is familiar with, giving him the chance to regain his form and show his potential.”
Verstappen has long been earmarked for a seat at Red Bull having impressed through the junior ranks and in his debut F1 season with Toro Rosso in 2015.
The 18-year-old became the youngest driver to score championship points in only his second F1 race of his career in Malaysia last year.
Verstappen has scored points in three of the four races this season with a best result of sixth at Bahrain.
The Dutchman's move is expected to create an eagerly anticipated intra-team rivalry with Ricciardo.
“The next step in my relatively short career so far, is an amazing opportunity,” said Verstappen.
“I really want to thank Red Bull Racing and Dr. Helmut Marko for the confidence they have in me.
“I'll have the chance to learn a lot from the top team that is Red Bull Racing.
“I'm also looking forward to working with an experienced and proven teammate like Daniel Ricciardo.
“Together with Red Bull Racing we'll do everything to prepare me as best as possible for my first laps in the RB12 Tag Heuer, next week in Barcelona. I cannot wait for that special moment to happen.”
Verstappen will make his first competitive outing for Red Bull in Friday practice for the Spanish Grand Prix held from May 13-15.