Valtteri Bottas claimed second, 14 seconds adrift of his team-mate; Kimi Raikkonen had been running second, but a puncture on lap 49 of 51 handed the position to Bottas and allowed Mercedes to extend its championship lead.
"It's been a great team effort in combination with the tyres".
Hamilton got a strong start to stay ahead of Raikkonen, whilst Vettel came under pressure from Verstappen as the pair switched places at Turn 1. With Lewis Hamilton running such a ideal race here in Great Britain, Kimi Raikkonen would have needed to get the jump on him off the line.
Teammate Kimi Raikkonen, who ran in second for most of the race, also sustained a puncture towards the end and did well to cross the line in third as Bottas benefited from Ferrari's significant misfortune to secure a Silver Arrows one-two.
Victory for Hamilton, whose 67th pole position put him just one short of Michael Schumacher's leading mark, came after a week in which he faced staunch criticism for being the only driver to skip a Formula One live event held on the streets of central London.
So now the tally stands: Vettel has 177 points in the season and is hanging on to a flimsy lead. "I think the same happened to him, but I don't know, I don't know what happened really". "So it was quite tricky - I tried everything". "I have more poles than most and I am building up the win". "I think we have a good vehicle, not fast enough for first, but for the race hopefully we can challenge them and have a good run".More news: Here's our first look at Atari's new console
"I think there's been a lot of growth with [team boss] Toto [Wolff] and I - from that meeting we had at the end of a year ago".
"Some people talk about the championship but it's still way too early and I'm still 60-70 points off Seb, so I am definitely not thinking about that".
With Hamilton pulling away, Vettel and Verstappen provided some fireworks with the Dutch teenager keeping the door firmly closed as they went wheel to wheel.
Force India's Esteban Ocon took eighth ahead of team-mate Sergio Perez and Williams' Felipe Massa completed the top-10 places. "So I tried to hang on to Lewis and tried to keep the gap but we just didn't have enough speed today". "It's just unbelievable what we've done today", Bottas said. After some expletives being shouted at in the Ferrari team radio, Raikkonen can swap his tires and rejoin in fourth place behind teammate Vettel.