And team boss Franz Tost said: "All things considered, the quality of the synergies with Red Bull Technology, the belief in our very high level of in-house technical knowledge, the impressive development trajectory Honda has shown so far, and the couple of strong young drivers we have available make me feel confident".
The senior Red Bull team will also be powered by Honda this year, opening up more opportunities for Toro Rosso to share certain parts under the FIA's "listed parts" regulations.
"I enjoyed my time working with Ferrari, which helped me keep in good shape and in touch with what is going on in Formula 1 on the technical front", said Kvyat, who tested Toro Rosso's 2018 vehicle during the post-race Abu Dhabi test.
The junior Red Bull squad finished ninth in the championship previous year, ahead of only the struggling Williams outfit.
The blue, silver and red vehicle, whose unveiling came after Haas F1's livery launch last week, reflected new aerodynamic restrictions with a simpler front wing and barge-board arrangement.
Toro Rosso finished ninth in the constructors' championship previous year and achieved a best result of fourth place in the Bahrain Grand Prix thanks to Pierre Gasly.
Tost said the new arrangement - which Toro Rosso last used in 2008, when it effectively had an entire vehicle from Red Bull before that practice was banned - would "leave more time for our technical working groups to focus more on the details".
Mickelson wins at Pebble Beach
All four players return to Pebble along with numerous game's best, including Jordan Spieth, Tony Finau and Jason Day. This was a fantastic win for the 27-year-old, a protege of fellow Aberdonian and former Open champion Paul Lawrie.
Toro Rosso will join the rest of the F1 teams at the preseason test in Barcelona on February 18.
"He had to live through a hard time when he joined us again in 2016 - it was psychologically tough, but no one has ever doubted his driving skills". "I believe in the guys in the team". However, the sidepod inlets have also been revised from last year's STR13 and now follow the design trend led by Ferrari in recent years.
"I don't want to say whether I've changed or not".
Kvyat's team mate is newcomer Alexander Albon, who finished third in Formula Two past year behind fellow 2019 F1 debutants George Russell and Lando Norris.
"My last race was 2017, then I was back a year after, pretty much".
"The way I've gone about my career and my racing is I never set any goals on anything". Just take it as it is.