Kvyat came through several fierce battles - most notably with Max Verstappen, the man who replaced him at Red Bull earlier this year - to clinch two points for his Italian team. It was the first time the Russian has qualified and finished inside the top 10 since this year's Chinese Grand Prix.
"I loved the racing today," Kvyat said afterwards. "To feel this passion again - I'm back to normal I think.
"All these months were a big test for my passion, and I'm loving it again. Pushing until the very last centimetre - it's mega."
Kvyat said his result could have been even stronger had the team taken more risks with strategy - or had he not lost two places while battling with Force India's Sergio Perez and Verstappen late on.
"Our strategy was a bit conservative," he reflected. "I'm happy because I took the maximum, but I feel we didn't try and take risks, and we were too conservative for our pace.
"I wanted to fight with [Fernando] Alonso [who finished seventh], which I think we could have done. But nevertheless it was a great result for the team, and really satisfying to be back in the points."
Commenting on his fight with Perez, Kvyat added: "I tried to pass Perez - it was very hard to pass, so I tried braking really late and ran all four wheels outside the line at Turn 7.
"Max also passed him [in the same move], and I had to let Perez past, so I had to let Max past too."
Kvyat's two-point haul strengthens Toro Rosso's hold on seventh in the constructors' championship: they now have 47 points, with Haas in eighth on 28. McLaren are sixth, seven points ahead of Toro Rosso.