Da Costa, Will Stevens and Roberto Gonzalez led all but 11 laps to secure victory in the blue riband round of the World Endurance Championship by 2m21.92s over the Prema entry shared by Robert Kubica, Louis Deletraz and Lorenzo Colombo, having at one point been a lap to the good.
The crew made up a lot of time during slow zones in the early phases of the race, with Da Costa acknowledging that "we got a little bit lucky in the beginning" and had built up a commanding lead as Da Costa and Stevens drove back-to-back during the night to allow the silver-rated Gonzalez to rest.
But da Costa revealed that both the #38 crew and its sister #28 car that finished third in the hands of Jonathan Aberdein, Oliver Rasmussen and Ed Jones started having tyre problems on Sunday morning, with da Costa reporting to his crew in the Porsche Curves near the end of the lap that he was experiencing a delamination in his third of a planned four stints.
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“I told the team in the last sector that I felt something strange in the tyre,” the Portuguese explained.
“And they told me, ‘You have to stay out, there is no driver ready to change.’ Because I could not do more time, I had to jump out.
“And I said ‘Guys, the tyre is destroying itself!’ They got Roberto ready in 20 seconds, and in the last moment I was coming past the pits, they said 'box!' and I just turned in to box.
“Honestly, we saved a lot of time here because the tyre was completely destroyed. To do a full lap like this would have been a big problem.
“We were the lucky ones.
"I felt it late, but at the same I don't give time to them to get ready, so it was a very hectic moment."
Da Costa estimated he lost two minutes to pitting under a safety car brought out when Robin Frijns crashed the #31 WRT entry at Indianapolis, when he was forced to wait at the end of the pitlane for the next safety car queue to pass him by.
“It made the race lively and made us stay honest the whole way,” he added.
“We had to add oil with three stops to go, I'm not sure if it was a precaution or not, and we had a lead so we did.
“And we started delaminating front-left tyres, us and the sister car. In my car two front lefts, the other car the same, it was not easy to keep that under control.”
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Prema had suffered a tyre failure itself with Deletraz at the wheel and reported overheating problems that blunted its challenge in the closing stages.
But da Costa said he couldn't be sure of taking the spoils until the flag after Kubica, Deletraz and Yifei Ye lost the LMP2 class win for WRT on the final lap last year.
“Honestly Le Mans chooses you, and what I've seen in the last few years with a Toyota stopping in the last lap [in 2016], last year the WRT stopping in the last lap, you just want to stay calm and composed until the car actually crosses the line,” he said.
"What a moment that was to be in the garage with all the mechanics, the families, our team, and seeing the car crossing the line was so special. I was crying like a baby!”
