Viviani outsprinted compatriot Alessandro Petacchi and 2013 Milan San Remo winner Gerald Ciolek at the end of the 209-kilometre, which lasted over six hours.
On the rising final kilometre through the Buccleuch Estate in Dumfries & Galloway, Viviani jumped clear of Ciolek in the final 150 metres, with Petacchi following him past for second on the road.
Victory gives Viviani a four second lead, thanks to time bonuses, at the top of the standings and the IG Gold Jersey of race leader heading into Stage Two, the Cumbria, the Lake District stage on Monday. The Cannondale rider also leads the Chain Reaction Cycles Points Classification.
"I am very happy for this win, but I hope to win again in the next few days” said Viviani speaking afterwards.
"This is my focus for the second part of the season so I hope for more in the closing part of the season. This year I make more times second place and this for a sprinter is no good.
"The finish was very hard because after six hours on the bike, the final kilometre was not flat so it was very difficult.
"I take the first position with my teammates with two kilometres to go, I go in the front and after I see what riders go to the sprint.
"I take the wheel of Petacchi and after Ciolek started his sprint, I take the wheel of Ciolek, because 200 metres after six hours is a very long sprint, so I wait one second and then I go.
"When Ciolek passed me I think 'Ok, 200 metres is so very long', so I take his wheel and pass him the last 150 metres.”
The race was shaped by a five rider break which went clear soon after the start in Peebles containing Kristian House, Ben Greenwood, Peter Hawkins, Aaron Gate and Christophe Laborie.
A maximum lead of just over five minutes was rapidly cut into by a Sky and Omega Pharma Quick-Step led peloton, and with the gap just under 30-seconds, Laborie's Sojasun teammate Anthony Delaplace jumped across, countering the break and going clear.
Delaplace's move with 44-kilometres to go earned him a lead of over a minute and a half, plus the Rouleur Combativity Award, but the Frenchman was reeled in on the first passage through the Drumlanrig Castle finish line.
With the peloton all back together Stage One came down to a bunch sprint, with the gently rising and curving final kilometre tailing off riders for a slow finale to an epic day.
Going into one of the hardest stages of The Tour, from Carlisle to Kendal and with further poor weather conditions predicted, Viviani said;
"I think for tomorrow that tonight I look at the roads, but for sure the time trial is not possible for me to keep the jersey. To take some more wins when it is possible is my aim, I think to win in London on Sunday is my second focus.”
Kristian House leads in the SKODA King of the Mountains Classification while AN Post Chain Reaction's Aaron Gate leads in the newly retitled Yodel Direct Sprints Classification, sponsored by parcel delivery company Yodel.
Will Stephenson (Rapha Condor JLT) finished in 87th place @ 01:40.
Highlights of Stage One are on ITV4 at 21:00 on Sunday evening, with Stage Two televised on both ITV4 and British Eurosport at 13:30 on Monday 16 September.
Stage Two see sees The Tour of Britain return to Carlisle for the start of The Tour's first ever visit to the heart of the Lake District. The 186.6-kilometre leg heads west to Britain's Energy Coast before turning into the heart of the Lakes and the SKODA King of the Mountains climb of Honister Pass.
After taking in Keswick and Windermere, Stage Two finishes in Kendal at Greenside atop the climb of Beast Banks, last used by the race in 2007.
Stage One Brief Result
1) Elia Viviani, ITA, Cannondale Pro Cycling
2) Alsandro Petacchi, ITA, Omega Pharma Quick-Step
3) Gerald Ciolek, GER, MTN Qhubeka
4) Marco Coledan, ITA, Bardiani Valvole CSF Inox
5) Matteo Pelucchi, ITA, IAM Cycling
6) Evaldas Siskevicius, LTU, Sojasun
7) Shane Archbold, NZL, AN Post Chain Reaction
8) Jon Dibben,GBR, Great Britain
9) Steele Von Hoff, AUS, Garmin Sharp
10) James Williamson, NZL, Node4 Giordana Racing
Chain Reaction Cycles Points Classification Leader: Elia Vivian, ITA, Cannondale
SKODA King of the Mountain Classification Leader: Kristian House, GBR, Rapha Condor JLT
Yodel Direct Sprints Classification Leader: Aaron Gate, NZL, AN Post Chain Reaction
Rouleur Combativity Award, Stage One: Anthony Delaplace, FRA, Sojasun
Under 23 General Classification Leader: Jon Dibben, GBR, Great Britain
For a full listings, vists the results page, here.