Sainz suffered a nasty shunt in Friday's second free practice session, crashing into the unprotected concrete wall on the outside of Turn 14.
The Spaniard managed to rebuild his confidence on Saturday and qualified second behind teammate Charles Leclerc.
From there he was forced to let Red Bull's eventual race winner Max Verstappen by at the start, but he then held off the second Red Bull of Sergio Perez, despite the Mexican enjoying at tyre advantage at the end of the race.
Sainz crossed the line in third some eight seconds behind Verstappen to take the final podium spot.
Speaking immediately after the inaugural Miami Grand Prix, Sainz revealed he was still feeling the consequences of Friday's heavy hit, admitting he was struggling with neck pain during Sunday's sweltering race.
"I've been better," he said.
"Obviously, after the crash from Friday I still had a bit of a neck pain going into the race, but I had to manage it and I fought through it.
"Especially with Checo on the end on the medium tyre, it was very difficult to keep him behind, but we managed to keep the podium, which is a decent result."
Even at the best of times Sunday's much anticipated event became an extremely physical challenge for the drivers due to the bumpy nature of the racetrack, and South Florida's heat and humidity.
Sainz, who is still looking for his first win, so far struggled to get on top of the Ferrari F1-75 and match teammate Leclerc, but at least ended a run of two consecutive DNFs with a third podium of his 2022 season.
"It wasn't easy at all," he acknowledged. "It's been a tough race with the tyres, with the heat. The car was moving and sliding a lot.
"But in the end, we got what we deserved I think, which is a decent P3, and we can build it up from here. I want more but it's not bad."
