The win helped Vettel increase his championship lead over Lewis Hamilton, who recovered from an opening-lap collision with Kimi Raikkonen to finish second. It soon developed into a Ferrari-Mercedes duel, with Raikkonen also challenging the third-placed Hamilton.
Hamilton's next deal - which could be announced in Germany in a fortnight - will see the Briton remain with Mercedes, a team where he has won three of his four world championships, for at least another two years.
Victory on Sunday would be Hamilton's fifth in a row at Silverstone.
Hamilton has promised he will "not give up" in what he describes as the "most intense battle we've probably all had" after the disappointment of not winning his home British Grand Prix.
Hamilton's team-mate Valtteri Bottas then led and a second safety vehicle, after Renault's Carlos Sainz and Haas' Romain Grosjean collided, kept the front four nose to tail and only fractions apart.
"Matteo Bonciani of the FIA said "be careful with him, he's really not happy at all at the moment".
"I had to go over the limit to get that time out of the vehicle and I could have easily not pulled that lap together, so I'm thankful that I was able to do it".
Hamilton's charge up the field proved that he is a brilliant driver, but also highlighted the difference between the engines in F1.
"I'm not the stewards, I can only think they have felt it was a little more serious than what Sebastian did in France", said Whiting.
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"I lost almost 3kg trying to get back to the top for you and my team", he wrote.
"But I believe it's nothing more than a racing incident".
"Once I was ahead, obviously I could use that advantage to pull out a gap and control the race from there".
"Obviously we're in a tough battle this year and it's the most intense battle that I think we've probably all had but I'm very, very grateful that the vehicle was still in one piece after the first turn three incident. Sometimes we say dumb s*** and learn from it". With Scuderia Ferrari driver Vettel in front, Mercedes made a decision to do the opposite of what Vettel did - ultimately choosing to leave him out, giving him track position but on older, slower tyres.
Lewis Hamilton fans out in force at Silverstone during 2018 British Grand Prix qualifying. Thank [sic] to everyone for the incredible support this weekend, we win and we lose together!
"The others pitted under the Safety Car, but we took the risk and stayed out to get ahead of Sebastian".
As for Mercedes' chances on Sunday, Wolff added: "We know already that it will be a tough race tomorrow, and especially so with two Ferraris starting right behind Lewis".
He had earlier told Sky Sports F1 that Mercedes technical director James Allison had asked "do you think it is deliberate or incompetence?" during the race.