The STR14 is the second Honda-powered auto the team have produced, and features a similar livery design to last year's challenger.
Toro Rosso's new auto features many parts that were used on the 2018 Red Bull.
Kvyat enjoyed a rapid run up the motorsport ladder, reaching F1 with Toro Rosso in 2014 before he was promoted the following year to Red Bull Racing.
"The chassis itself, as per regulations, is a complete Toro Rosso design and IP as is the front outboard suspension, along with all aspects of the cooling systems, power unit installation and steering column".
"The aim is to always improve in order to compete at the highest possible level and this winter we have worked intensively to bring a few organisational changes, which we hope will help smooth a few issues we experienced in the past", said team principal Franz Tost.
The team have a new driver line-up with the return of Russian Daniil Kvyat and a debut for UK-born Thai Alexander Albon. Daniil Kvyat first drove for the team five years ago and is making his return after spending 2018 as one of Ferrari's simulator drivers.
"We don't know exactly where we're going to be", he admits.
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For instance, the entire Honda-powered rear end of the new Toro Rosso will be identical to what is run by Red Bull.
"If I was a dictator, which I never was, I would take the FOM money and give - to just name a figure - $60 million to each team". Me and my engineering crew will do everything we can to squeeze the maximum out of the package we're given by the team.
There is also an increase in personnel at Honda's research and development facilities in Sakura and Milton Keynes to develop and prepare twice the number of engines, and Honda has "strengthened the functionality" of its mission control rooms at those locations to deal with working for two teams on race weekends.
Toro Rosso finished ninth in the constructors' championship a year ago and achieved a best result of fourth place in the Bahrain Grand Prix thanks to Pierre Gasly.
Kvyat and Albon replace Pierre Gasly, who has left for Red Bull, and Brendon Hartley, dropped after a disappointing year.
"Given we have new people on board, we need to do our best to integrate them within the team and find the best practices to work efficiently with each other".