The pair will be in their new positions until the end of the season.
Last year teenager Verstappen became the youngest driver in the history of F1 at the age of 17.
Red Bull team principal Christian Horner described the 18-year-old Dutch driver as "an outstanding young talent" and said he was "pleased to give him the opportunity to drive for Red Bull Racing".
It is a typically ruthless decision from Red Bull, who use Toro Rosso to test out young drivers and then either promote them within two years or dump them.
And it raises serious questions over the F1 future of 22-year-old Kvyat, who was described by Red Bull team boss Christian Horner as "the real deal" when he was promoted to replace Vettel in 2015 after just one year at Toro Rosso.
Horner said: "Dany will be 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."
The Russian, who has contested more grands prix for the senior team than the junior one, scored more points than team-mate Daniel Ricciardo in his first season at Red Bull but has been out-paced more often than not by the Australian.
Kvyat has lagged behind Ricciardo in qualifying at every race so far this season, although he scored the team`s only podium finish of the season at the Chinese Grand Prix after Ricciardo suffered an early puncture while leading.
Kvyat has been involved in controversies with Vettel at the last two races.
In China, Vettel confronted him following the race after Kvyat made an aggressive overtaking move at the first corner that resulted in the German colliding with Ferrari team-mate Kimi Raikkonen.
Most observers felt Kvyat had done nothing wrong in that incident, but the same could not be said of the collisions between the Red Bull driver and Vettel in Russia last Sunday, the second of which resulted in the Ferrari driver`s retirement.
Kvyat crashed into the back of Vettel twice in the space of a few hundred yards, prompting the four-time world champion to launch a string of expletives over his team radio and then to speak to Horner about Kvyat`s driving.
Verstappen`s stellar debut season put him on the radar of the sport`s two top teams, Mercedes and Ferrari, both of whom have considered him as a potential driver for 2017.
But he is under contract to Red Bull until the end of 2017 and the decision to promote him could be motivated at least in part by a desire to ensure they keep hold of arguably the hottest rising star in the sport.
