Kvyat was involved in a spat with Ferrari's Sebastian Vettel in Shanghai on Sunday after the German collided with team-mate Kimi Raikkonen at the first corner while the Red Bull steamed through on the inside.
Vettel accused the Russian of driving like a madman, saying he launched his car like a torpedo, but the result of the vote on the official www.formula1.com website left no doubt about who won the argument.
The driver award is new for this season, with an online vote through the website after each race, as a move to increase fan engagement in a sport that has seen declining television audiences.
With perfect timing, ahead of next week's Russian Grand Prix on the May Day holiday in Sochi, Kvyat took the second podium finish of his career and delighted his bosses.
Team principal Christian Horner, whose bid to secure a Ferrari engine was rebuffed last year, rubbed salt in his former driver Vettel's wounds by saying Kvyat had put in a "very measured drive".
Vettel won four world championships with Red Bull between 2010 and 2013 and finished second in China but 37 seconds behind Mercedes winner Nico Rosberg, who is now on a six-race winning streak.
"Mercedes are still a step ahead, but we're snapping at the ankles of Ferrari," Horner told reporters.
Australian Daniel Ricciardo had started on the front row, ahead of Vettel and Raikkonen, but his hopes were wrecked by a puncture on the second lap when he ran over debris.
Ricciardo still finished fourth, for the third race in a row, with Raikkonen fifth.
He said the puncture and a further setback caused by the deployment of the safety car, which left him with even more places to make up, was "like getting punched in the stomach by a heavyweight".
He added however that his recovery had been one of the best races of his life and saw plenty of reasons to be optimistic about the future.
"It’s really promising for sure. Right now three races in, every weekend we’ve shown a strength at some point and it’s been a strength we didn’t think we would have at this point in the year," he said.
Red Bull failed to win a race in a bad-tempered 2015 overshadowed by a rift, later patched up, between them and engine partners Renault over the lack of performance from the power unit.