As he arrived in parc ferme following his third-place finish, Kvyat revealed his partner Kelly Piquet -- the daughter of three-time world champion Nelson Piquet -- gave birth to a baby girl Saturday night.
It is the couple's first child and was celebrated in style by Kvyat by taking a surprise podium at Hockenheim.
Speaking after the race, he dedicated his result to Piquet and his new daughter.
Kvyat's podium was his first since the 2016 Chinese Grand Prix, which he scored with Red Bull two races before being dropped by the team. After a year on the sidelines last year, Kvyat got a surprise call up by Toro Rosso ahead of the 2019 season and the podium is the team's first since Sebastian Vettel won the 2008 Italian Grand Prix.
"It was amazing," Kvyat said. "Amazing to be back on the podium, incredible for Toro Rosso after so many years, to bring a podium to the team is amazing.
"The race was crazy. Finally I managed to put everything together to get this podium. I'm really happy.
"It was a horror movie with a black comedy. At some point I thought the race was done for me but it came alive again. It was incredible.
"It was incredible. A rollercoaster. A bit like my whole career."