Stannard won by nearly two minutes from breakaway companions Graham Briggs (JLT Condor) and Kristian House (One Pro Cycling), to lead a British 1-2-3 at the end of the 180km stage from Congleton, with the peloton racing in over five and a half minutes down.
"It's a pretty big day for me. I was motivated for this stage because it was pretty much on home roads," Stannard said.
"I was keen to get away. It's always nice to win from a breakaway and go solo too. I really enjoyed it today.
"I knew I want to go over the top of the Cat and Fiddle with about five minutes on the peloton, we'd have a chance on the run-in then. I heard it went out but I didn't know much more. I was just happy to be up there."
Overnight leader Vermote retained his six-second lead over Steve Cummings (Dimension Data), with the other main general classification positions remaining unchanged.
"It was a hard parcours, but we took it easy, because the riders in the escape were a long way down in the standings", Vermote said.
"I enjoyed this day. Not only wearing yellow was great, but also having all these incredible fans cheering us from start to finish.
"Concerning tomorrow, it's difficult to say what will happen in that long and challenging stage, but what's certain is that we will try to defend the jersey."
Matt Cronshaw of Madison Genesis attacked as soon as the race was de-neutralised on the outskirts of town, taking Briggs, House and Stannard with him.
The break was down to three by the time it reached the second climb, with Cronshaw distanced and eventually swept up on the day's final categorised climb of the Cat & Fiddle.
Over the top of that climb as the route navigated a series of false flats across the moors of the Peak District National Park Stannard went away, casting Briggs and House adrift to then contest their own race for second.
Bardiani CSF's Nicola Ruffoni led in the peloton home for fourth, outsprinting Danny Van Poppel (Sky) and Ramon Sinkeldam (Giant-Alpecin), who claimed the points jersey.
Andre Greipel (Lotto-Soudal) holds the sprint jersey going into Stage Four in Wales, while Xandro Meurisse (Wanty Group-Gobert) keeps mountains classification jersey.
The Belgian enjoyed a day long tussle with Nicolas Roche for the remaining points at each of the three climbs, but only managed to extend his hold on the jersey by a solitary point.