The 188-km ride from Edinburgh was decided in the final few metres as Orica Scott's Ewan, given the perfect lead out on the cobbles, just got his wheel in front.
Team Dimension Data's Norwegian Boasson Hagen, a former winner of the British race, took second place ahead of Team Sky's Olympic track champion Viviani.
Eight riders had formed an early escape, putting three minutes between themselves and the peloton.
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Poles Karol Domagalski and Lukasz Owsian were the last of the leaders to stay ahead but they were reeled in on the final climb of the day, after which it was all about the sprinters.
"When I crossed the line I was pretty confident that I got it," Ewan told the Tour's website. "I know it was pretty close, but I thought I had got it.
"Then I saw all the cameras go around Eddy and I thought hang on a minute. When it is that close you just don't know, and I saw Viviani as well round the outside. I was definitely happy to learn that I had won."
Monday's 211-km stage begins in the Kielder Forest in Northumberland and heads towards the east coast of England, finishing at Blyth.
(Reporting by Martyn Herman; Editing by Clare Fallon)
