The 19-year-old Russian completed 22 laps for Toro Rosso at Silverstone on Friday and was 11th fastest. He lost some track time with a spin.
Asked by AUTOSPORT if he felt the test had brought him closer to an F1 seat, Kvyat replied: "Today made my dream even bigger.
"My first dream was to drive an F1 car. Today it's done, and now after this session I feel like with more mileage I should get more comfortable with the car and I'm confident I could be quick in this car.
"Unfortunately it's really limited for young drivers these days but hopefully my job has not gone unnoticed. I'm sure it hasn't.
"For now, it's not up to me. My main concern is the championships I'm racing in.
"I did the best I could today. We had a few issues, but I think the team saw what they needed to from me.
"I need a bit more time to analyse what happened today. I can't tell if I'm ready [for F1] next year, tomorrow, or whenever.
"For sure one day I want to be in F1 - today I understood this."
The driver line-ups in Red Bull's F1 teams are set for changes next year with Mark Webber's exit for the World Endurance Championship raising the possibility of Daniel Ricciardo graduating from Toro Rosso to Red Bull.
Along with Antonio Felix da Costa and Carlos Sainz Jr, Kvyat is a leading light of Red Bull's junior programme in the F1 feeder categories, but said he was not watching the situation too avidly.
"I'm not giving it too much attention at the moment," said Kvyat, who currently races in both GP3 and the European Formula 3 Championship.
"It's not really my business so far, but of course I keep an eye on it."