The opening laps were run behind the safety car after a heavy downpour shortly before the start, but once it had peeled off it was non-stop action as the track gradually dried, prompting the field to switch first from wet to intermediate tyres, and then to slick rubber from around one-third distance.
As Hamilton controlled from the front, Rosberg and Verstappen were locked in battle for much off the afternoon, both men pulling off superb passes on the other. Barely a second separated them at the flag, as Rosberg held on despite late transmission issues which required him to avoid using seventh gear - although the positions were subsequently reversed.
Daniel Ricciardo was a distant fourth in the second Red Bull, with Kimi Raikkonen an even more distant fifth in what was a dismal afternoon for Ferrari. Team mate Sebastian Vettel was ninth after picking up a penalty for forcing another car off track.
The Force Indias of Sergio Perez and Nico Hulkenberg took sixth and seventh places, with the top ten completed by the Toro Rossos of Carlos Sainz and Daniil Kvyat in eighth and tenth respectively.
After the heavens had opened 15 minutes before the start, and the first five laps run behind the safety car, the track was ready for intermediate Pirelli tyres in place of the mandatory wets on which everyone had started.
Once the wholesale pit stopping was over, and the racing began again after a brief Virtual Safety Car deployment on Lap 7 as Pascal Wehrlein aquaplaned his Manor off into retirement, Hamilton was 4.8s ahead of Rosberg, who was initially struggling for a while in the slippery conditions. For the next eight laps the Silver Arrows were running one-two, until Rosberg made a mistake and the ever-threatening Verstappen went round the outside of him in a brave move at Chapel to snatch second place.
As Hamilton continued to lead after the next round of pits stops saw everyone switching to the medium Pirellis, Rosberg had a real battle on his hands to repass Verstappen, and that’s where he lost his chance of victory. The Dutchman not only stayed in front for lap after lap, but kept Hamilton honest at the same time. Both he and Hamilton had lurid off-track moments at Abbey and Farm, as did many others as the track remained greasy, but time and again the Red Bull driver resisted the second Mercedes at Stowe.
Eventually Rosberg found a way by there on the 38th lap, just as a very distant Vettel, down in ninth, eased Felipe Massa’s Williams off track and earned himself a five second time penalty.
Finally freed to push hard, Rosberg then launched a flurry of fastest laps as he cut the gap from 8.7s to 6.1s by lap 45. But then on the 46th lap he reported gearbox problems and was instructed by his crew to change through sixth to eighth gears, without using seventh. Remarkably, he still managed to close the gap to Hamilton, who was simply doing what he had to do to manage his race. By the finish the gap between the two Mercedes was 6.9s.
However, Rosberg's subsequent 10-second post-race penalty - which Mercedes intend to appeal - dropped him behind Verstappen, who was just 1.3s behind at the flag. The latter’s performance was hugely popular, and he finished 17.9s ahead of team mate Ricciardo.
Behind them, a terrible day for Ferrari yielded a best finish of only fifth for Raikkonen who overtook Perez’s well-driven Force India with six laps to run. By the flag, the Mexican had team mate Hulkenberg right on his tail, as Sainz gave Toro Rosso more points with eighth. Vettel retained ninth place by 0.9s from Kvyat once his penalty was applied.
It was a tough day for Williams and McLaren too, with Massa 11th from Jenson Button, and Fernando Alonso 13th after a huge, high-speed spin through the gravel at Abbey, ahead of Valtteri Bottas.
Both Manors spun off and both Renaults retired, with Jolyon Palmer having picked up a stop-go penalty after his crew released him from a pit stop with his right-rear wheel not properly attached.
Sauber’s Felipe Nasr and Haas’s Esteban Gutierrez took 15th and 16th as their respective team mates Marcus Ericsson and Romain Grosjean failed to go the distance.
Hamilton’s hat-trick - his 47th F1 win - makes him the first three-in-a-row British Grand Prix winner since the legendary Jim Clark in the Sixties, and the gap to points leader Rosberg is now down to just one point - 168 to 167 - with the notional seasonal midpoint coming up in Hungary in a fortnight.