After rain delayed the start of the race by more than 90 minutes, Hamilton was one of the few drivers who switched from intermediate tyres to slicks on a track that was still drying.
The Briton had a slow start and then found himself going backwards, falling as low as 18th, but he opted to stick with his strategy despite Ferrari calling him back into the pits for a tyre change.
Hamilton then carved his way through the field as the track dried up while safety cars also came to his rescue, with the veteran eventually overtaking Red Bull's Isack Hadjar in the dying stages to secure a podium behind Max Verstappen and Kimi Antonelli.
"As a team, we ended up taking that soft (tyre) and at the end of the pit lane, these two (Verstappen and Antonelli) took off their (tyre) blankets and my heart just sank, because I knew straight away that we were in trouble," Hamilton told reporters.
"As we pulled out, all the spray came out the back of Max's car and I was like, 'Okay, I'm done. My race is really in trouble'.
"We made the decision to live with it. As everyone was overtaking me, they (Ferrari) said, 'Should we stop?' I said, 'No, we've already waited another 26 seconds. What's the difference?"
But as the track dried and safety car deployments played into his hands, Hamilton began a remarkable recovery drive as he gave Hadjar chase for fourth place, before George Russell's retirement gave him a shot at a podium.
"I just waited for the track to dry and once it did, it felt good that I was catching people. The car felt great. Then after that, really good pit stops from my guys in the garage," Hamilton added.
"And then a good strategy. So for us to get back to the podium, I never thought that that's what was going to be the case. I'm really, really grateful."
His teammate Charles Leclerc also salvaged fourth place for Ferrari and he conceded that was the best they could have achieved considering how much faster Red Bull and Mercedes were.
"Honestly, the first part of the race, everything that could have gone wrong went wrong. And on the second part, everything that could have gone right went our way," Leclerc said.
"So at the end of the day, I prefer it that way than the other way around. We were P4 today and I honestly think that with a normal race we would have finished P3 and P4. The two guys in front were just way too quick for us.
"Lewis had quite a good pace. I struggled a lot with the car, which we're looking into. I had a good feeling on Friday, good feeling until Q2, and then Q3 I started to lose out. Today the race was very tricky."