Red Bull team principal Christian Horner said the switch would allow Verstappen to showcase his ‘outstanding' talent, and also help Kvyat ‘regain his form'.
The move comes just days after Kvyat was heavily criticised - by Horner included - after twice coming together with Ferrari's Sebastian Vettel in Russia.
"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," Horner said.
"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."
Kvyat was promoted to Red Bull ahead of the 2015 season, in place of the departing Vettel. He out-scored team mate Daniel Ricciardo in his maiden season with the team, by 95 points to 92. This year the Russian secured the second podium of his F1 career in China, but then cost the team a chance to score points with his double-collision with Vettel.
Verstappen, meanwhile, is widely regarded as one of the best talents on the grid. The youngest driver to compete in the sport, and the youngest points-scorer, he twice came close to scoring a podium during a superb debut campaign with Toro Rosso last year.