Double Olympic champion Robert Scheidt of Brazil and French triple Olympian Jean Baptiste Bernaz both needed Meech to finish well down in the 10-boat final to have any chance of grabbing bronze themselves.
"I saw them slowing down and waiting for me, they were trying to slow me down to get me back in the fleet," Meech said.
"So that was a little scary. But eventually they had to worry about each other too so I managed to survive."
The race was won by Scheidt, who finished four points behind Meech in fourth.
Australian Tom Burton finished third to snatch gold from Croatian Tonci Stipanovic, who limped across the line second-last after a pre-start joust with Burton backfired.
Meech was focused on trying to stay out of trouble, reading the wind shifts well to establish a good early position.
It mirrored a campaign in which he was consistent before powering into contention with victory in the 10th and final fleet race.
The hardest part was sitting through still conditions on Monday, when the medal race was eventually postponed for a day.
"I really wished we could have raced yesterday, I was pretty nervous. It was hard to go home last night but that's sailing, we wait around a lot. I've been doing it for the last 5-10 years."
Meech's sister Molly is competing in the 49erFX class and sits within one point of the lead heading into the medal race on Friday morning. Her crewmate is Alex Maloney.
Sam Meech says he will support his sister but won't give her any advice, confident she can finish higher than him on the dais.
-NZN
