The problem dropped Ferrari's Formula One championship leader from third to seventh, slashing his advantage over Mercedes' Lewis Hamilton to a single point with half the season remaining.
"A full investigation has now confirmed that the original cause of the failure was a slow puncture," Pirelli said in a statement.
"The consequent driving back to the pits on an underinflated and then flat tyre led to the final failure."
Vettel, whose set of soft tyres had done more than half the race distance, had said on Sunday that the failure was "quite sudden".
"The fact is that the tyre blow-ups came by surprise. It's not like we were gambling and jeopardising our race result," he had said.
His Finnish team mate Kimi Raikkonen, who finished third at Silvesrtone, also had a problem with the front left tyre in the closing laps and had to pit for a fresh set.
Pirelli said the Italian company was still investigating the cause of that failure.
(Reporting by Alan Baldwin, editing by Pritha Sarkar)
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