Bottas Mercedes seemed to run over some debris on the main straight, which lead to his right-rear tyre getting punctured and gave Hamilton the race on a platter.
The record books will show that the 2018 Azerbaijan Grand Prix was won by Lewis Hamilton - hist first win of 2018 - that Kimi Raikkonen grabbed second place and Sergio Perez got a podium for Force India in third.
After the restart, the German made a mistake and dropped to fifth place.
With the safety vehicle in, Vettel immediately set about overtaking the leader but he locked up and ran wide, dropping to fourth behind Hamilton and teammate Kimi Raikkonen.
Hamilton said in his interview that the Azerbaijan Grand Prix was an emotional race.More news: Demonstrations resume in Armenia following opposition leader's call
"Apart from that, I think it was a very good race; we had a good pace and were in control".
Fernando Alonso finished seventh for McLaren ahead of Canadian Lance Stroll of Williams, Stoffel Vandoorne in the second McLaren and New Zealand's Brendon Hartley of Toro Rosso who scored his first point in F1.
Hamilton criticised race director Charlie Whiting for not stopping the race, rather than sending out a flat-bed truck to recover Grosjean's auto.
It was a race where the safety vehicle once more played a massive role, with series of crashes.More news: Lea Michele and Zandy Reich Are Engaged
A frustrated Hamilton complained about back markers, but began clawing back time, recorded a fastest lap and then saw Bottas go eight-tenths quicker. "There was even a tiny chance to win the race on a normal strategy as well".
It was fitting that the race that started with a bit of collision and mayhem at the beginning with Raikkonen and Force India-Mercedes' Esteban Ocon colliding (among others) ended on a similarly unpredictable note.
Far tamer racing ensued after the safety auto came in on lap six.
Ferrari's Sebastial Vettel, who had led the championship before Baku and looked comfortable for his 50th career win and third out of four this season, finished fourth.
That kept the safety vehicle out for even longer and when racing resumed Vettel passed Bottas but then ran wide as Hamilton and Raikkonen swept through. Sebastian eventually followed suit on Lap 30, but Valtteri stayed out - Mercedes figured his worst result would be third but that a well-timed safety auto would allow him to make a free pit stop and keep his lead from the Ferrari. The other Mercedes of Valtteri Bottas, however, kept circulating in a bid to stay out as long as possible and at least try to come out ahead of Hamilton.More news: 'Schoolhouse Rock' founder Bob Dorough dead at 94
The endeavour from Carlos Sainz to get between the two Red Bulls early on and get the Renault works team in contention was fantastic - taking advantage of the duelling team-mates and dividing the pair.
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