Red Bull stunned F1 on Monday night by announcing that the 19-year-old had been picked ahead of Antonio Felix da Costa to replace Daniel Ricciardo.
Kvyat drove for Toro Rosso in the Silverstone young driver test in July, and carried out a straightline run for Red Bull last week.
"Daniil is a very talented driver, which can be seen from his results in all the junior categories," said Tost.
"He impressed our team with a strong performance and very informative technical feedback at the young driver test in Silverstone.
"This suggests that the basic qualities from which he can progress are all in place.
"Daniil can be sure that we will use all our experience of training youngsters to give him the best possible start to his Formula 1 career."
Kvyat declared himself thrilled at the news of his surprise promotion.
"This is fantastic news, a dream come true and I want to thank Red Bull and Toro Rosso for giving me this priceless opportunity," he said.
"Ever since I began karting, I wanted to get to Formula 1 and now that wish will become reality next season.
"I had a brief taste of working with the Toro Rosso team, when I drove for them at the Silverstone test and I enjoyed the experience very much.
"The fact I am based in Italy and speak Italian will, I am sure, help me to become part of the team very quickly."