The 20-year-old Russian has made a big impression in his first four grand prix outings, scoring points on three occasions and generally showing better than his more experienced team-mate, Jean-Eric Vergne.
And Tost - who oversaw Sebastian Vettel’s first steps in Formula 1 during 2007 - believes his latest protege can also make it to the top of the sport.
“Daniil has done a really good job so far,” Tost told the official Formula 1 website. “He has shown that he is an extraordinarily skilled driver with a very high natural speed.
“He’s shown [his potential to win titles] already by winning the younger classes - and especially the way he was winning. He has it in his DNA to win races, and championships.
“There is still potential to improve - and he is at the beginning of the learning process. Coming into F1 is one thing, but then the crucial question is how and when the curve runs north - and his curve is definitely running north!
"If we provide him with a good car he will finish some races within the top five or six drivers this year," predicted Tost.
Kvyat became the youngest man in F1 history to score points when he took ninth place at the season-opening Australian Grand Prix, breaking the record set by Vettel seven years ago. Two more top-10s in Malaysia and China have cemented his status as the most impressive rookie on the 2014 F1 grid, and led to comparisons with four-time world champion Vettel.
Kvyat and the Toro Rosso squad will be targeting further points in this weekend’s Spanish Grand Prix at Circuit de Catalunya.