Kvyat misjudged his braking point heading down into Turn 2, and lightly tagged the rear of Vettel's car - pitching the Ferrari into Ricciardo. All three recovered, but as Vettel headed through Turn 3 Kvyat tapped his rear again, this time diverting the Ferrari into the barriers and out of the race.
The incident came just one race after Vettel and Kvyat's war of words following the Russian's aggressive start in China.
"There was a bit going on [at the start]," Kvyat explained, "and I think in the last moment I didn't expect it to slow so much.
"I tried to press the brakes, the rear wheels locked and the first contact came from that. The second contact he just slowed down a lot, I couldn't see ahead, and I didn't have time to react.
"All the mess came from me… of course it doesn't feel great but these things happen sometimes. It's probably the messiest [start] of my career. I will learn from it - and of course apologies to everyone involved.
"I think we will speak [with Vettel and Ricciardo], that's all we can do. Everyone will attack me, but I'm okay with that."
Vettel was incandescent in the cockpit of his Ferrari immediately after the contact, but was more circumspect in the paddock.
"I got a big hit, I was lucky not to spin and then got another huge hit which turned me around and I couldn't avoid the barriers," he said.
"These things happen - there is nothing I could have done differently. I think there was enough damage to conclude I couldn't continue. If anyone needs to talk to anyone, I think it's him [Kvyat]…"
Red Bull team principal Christian Horner, who discussed the incident with Vettel during the race, added: "Obviously Seb was slightly frustrated. All I could do was apologise, because this week it was unfortunately a mistake from Dany.
"I think he knows what happened. He's misjudged that first corner, and hit Sebastian who hit Daniel. From a team point of view it's screwed our race because we could have scored a lot of points today.
"Emotions run high with all these guys, in his home race I think he just went for too much too soon. Unfortunately I think he has just misjudged it."
Kvyat, who received a 10-second stop and go penalty and three penalty points as punishement for the incident, eventually finished a lap down 15th place.
WATCH: Vettel crashes out in dramatic Sochi start
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