Kvyat had been fighting with Sebastian Vettel in the closing stages of the Grand Prix, at one point closing to within a few tenths of a second of the Ferrari driver.
Vettel made a crucial call to pit on lap 43, however, while Kvyat stayed out one lap longer - which proved doubly costly as he spun on the next lap. Without those two setbacks, Horner said the Russian might have had a shot at the top three.
"Dany drove a great race, and all weekend he has done a good job," Horner reflected.
"He possibly could have been in contention for a podium if we had pitted him earlier when he was fighting with Sebastian, and he also spun on his in-lap - but nonetheless he drove a very strong race."
Kvyat said could draw comfort from the fact Red Bull made progress over the weekend after introducing several upgrades for the RB11.
"The team have done a great job and the car is handling really well," he commented.
"We were really competitive today, the race was really fun, a little crazy with the rain, and I was enjoying myself. We will keep pushing to improve and move forward over the next few races."
Kvyat's eight-point haul was enough to keep him just ahead of Force India's Nico Hulkenberg in the fight for eighth place in the drivers' standings.
