Ricciardo has been holding off Vettel for fifth position before the incident occurred.
Kvyat went on to make further contact with Vettel which sent the Ferrari into the barriers at Turn 3.
The Russian received a 10 second stop/go penalty and three points on his licence.
Following the incident, Ricciardo recovered to the pits where the team fitted medium tyres.
Unfortunately the tyres were unable to complete the remaining race distance which saw the Western Australian finish 11th after a second stop for fresh rubber.
“It all ended at the start. Obviously we finished the race but we had a lot of damage from the second corner,” said Ricciardo.
“We don't really know what our real pace was. We tried to do something but there was a lot of damage.
“I saw Seb come into me but I knew it wasn't him (that made a mistake) because the way he hit me was not natural. I knew he had got hit from behind.
“It is frustrating. Obviously not only my race but a few were ruined today. It is just a shame that it was over really by the first lap.
“Having a small look at the replay, Seb was on my inside, I was in the middle and Sergio (Perez) was on the outside so there were three cars taking up the whole track.
“If Dany hits Seb I don't know where he was going. It was not like there was room on the inside.”
Ricciardo has dropped to fourth in the championship standing after failing to land a points finish at Sochi.