Hamilton led from start to finish to win from his Mercedes teammate Valtteri Bottas, but Vettel, running third at the time, saw his left front tire fail with just over a lap of the 51-lap race remaining.
The German limped back to the pits but had slipped to seventh place when he emerged.
The same fate had already befallen Vettel's Ferrari teammate Kimi Raikkonen, but the Finn was able to salvage the final spot on the podium.
After his recent disappointments, Hamilton put on a dominant display in front of his British fans, drawing him level with compatriot Jim Clark and Frenchman Alain Prost for victories at Silverstone.
It was his fourth straight victory in his home grand prix and never in doubt once he had pulled clear of Raikkonen from pole position at the start.
Raikkonen ran in a distant second for much of the race until his was the first of the Ferraris to suffer an unexpected tire problem.
"Mercedes were a bit faster than us today," said the Finn. 'Unfortunately, unlucky situations seem to be following us at the moment," he added.
Bottas, charging through from ninth on the grid after a penalty for a gearbox change, overtook Vettel on his way to an excellent second place, while the Red Bulls of Max Verstappen and Daniel Ricciardo moved up to fourth and fifth in the late drama.
Vettel's misfortune could prove a key moment in the title race, with Hamilton after his fourth win of the season, clearly delighted by the outcome.
"I'm so proud I could do this for you," he told the crowd at the trophy presentation.
"Valtteri had an incredible drive to go from ninth.
"Now the plan is to go for the championship."
It was the 57th career win for the 32-year-old Hamilton, who had started the weekend 20 points adrift of his arch-rival Vettel after a disappointing fourth place in the Austrian GP won by the improving Bottas.
Vettel, now under intense pressure ahead of the next round in Hungary in a fortnight, could not hide his disappointment at the late problem. "It was quite sad, I don't have a lot to say," he told the team's official twitter feed.
"I was pretty sure the tires would be fine."