Hamilton was relieved after escaping action for allegedly blocking Romain Grosjean during qualifying, with stewards ruling that the Haas car was not impeded.
Hamilton sped around Silverstone in 1 minute, 26.6 seconds in front of his home fans. They cheered the three-time Formula One world champion at the end of a week when he faced criticism for being the only driver to snub a promotional event in London.
"I feel amazing, especially with a great crowd like this around the track," Hamilton said.
"These typical English conditions are what we grew up racing in. I'm comfortable in it."
Hamilton, who is 20 points behind leader Vettel in the Drivers' Championship, is now one short of Michael Schumacher's all-time record 68 poles.
Valtteri Bottas was fourth fastest, but, like Red Bull's Daniel Ricciardo, he has a five-place grid penalty for Sunday's race after being forced to change the gearbox in his Mercedes.