The Russian racer rammed into the Ferrari of former Red Bull hero, four-time world champion German Sebastian Vettel, twice on the opening lap of an eventful race won by Nico Rosberg of Mercedes.
Vettel and two other drivers were forced to retire on the opening lap and both Kvyat and his Red Bull team-mate Australian Daniel Ricciardo's cars suffered severe damage.
They continued, but finished out of the points adrift of victorious German Nico Rosberg who won ahead of his Mercedes team-mate defending champion Briton Lewis Hamilton. Rosberg now leads Hamilton by 43 points in the title race.
It was Rosberg's fourth straight win this season and seventh in succession, including three last year, and was a disastrous day for the chasing pack including the Ferrari and Red Bull teams.
Red Bull boss Christian Horner apologised to Vettel and Ferrari for Kvyat's impetuous driving in Turns Two and Three and the Russian afterwards apologised to Vettel and 'everyone involved' for his reckless moves.
After the race, Red Bull's motorsport advisor Austrian Helmut Marko, the hard-headed consultant who oversees the team for Red Bull owner Dieter Mateschitz, made clear his displeasure.
“Let's put it this way,” he said. “Kvyat was over-motivated. Braking too late once and hitting another car would be acceptable in front of his home crowd, but this doesn't apply for the second crash.
“It's a pity because he not only ruined Vettel's race - and Ricciardo's race - but also his own race. He had to do an extra stop. It was a day of disaster for Red Bull.”
It was the second successive race in which Kvyat had crashed into Vettel, having angered the German by careering into his car the start of the Chinese Grand Prix two weeks ealier.
Marko added that, contrary to the collision in Shanghai where he felt Kvyat's aggression had been acceptable, he said: “This time, in contrast to Shanghai, I understand him (Vettel) completely - he is right to be so angry.”
He said the team did not want to discuss the matter in detail in Sochi because emotions were too high. Ricciardo had demanded an apology from Kvyat after the race and the Russian said he spoke to Vettel by phone.
“Let's cool down a bit,” said Marko. “During the week, we will talk about this.”
Although Finn Kimi Raikkonen finished third for Ferrari, it was only a consolation to the team after the loss of Vettel in an atmosphere of rising pressure as they labour to keep pace with Mercedes.
Red Bull driver changes?
Red Bull has also made it clear that the team has ambitions to close the gap and revive its former glories and it has hinted that driver changes are likely.
The huge potential of Dutch teenager Max Verstappen and his team-mate Spaniard Carlos Sainz at the Red Bull 'sister' and 'feeder' team Toro Rosso are under close observation with a view to one of them stepping up - as both Ricciardo and Kvyat did.
The competitive tensions between all four drivers are intense and have become so strong at Toro Rosso that the team recruited the vastly-experienced former Manor team boss John Booth to assist in their management as a consultant director of racing.
Verstappen, 18, has been linked with Ferrari while Sainz, 21, is also regarded as a 'hot shoe' with a big future ahead of him.
Both are the sons of former racing drivers - Verstappen's father Jos raced in F1 while Sainz's father Carlos was a two-time world champion rally driver.
Original source: Kvyat's future with Red Bull in doubt