Wiggins was the bookmakers' odds-on favourite after his historic summer triumphs.
Accepting his award from the Duchess Of Cambridge, the 32-year-old said: "I'm not going to swear tonight, so I'm just going to say thank you very much to everyone."
He also raised a laugh when he joked about his grandmother rigging the vote. He said: "The cheque is in the post because you pressed redial God-knows how many times."
Olympic heptathlon champion Jessica Ennis was awarded second place at the event at London's ExCel centre, while Scots tennis star Andy Murray took third place for his US Open victory and runners-up spot at Wimbledon.
The Duchess, who was on her first public Royal engagement following her spell in hospital for severe pregnancy sickness, was presenting the award in her role as an official ambassador of Britain's Olympic and Paralympic team.
She did not speak while presenting the award and looked on as Wiggins told his children Ben and Bella "you can go to bed now".
Glasgow-born rower Katherine Grainger was among the Scots shortlisted for the award after the 37-year-old took gold in the double sculls with Anna Watkins at the London Olympics.
Sir Chris Hoy was also nominated after a year that saw the cyclist take his tally of Olympic golds to six.
Olympic and Paralympic champions dominated this year's Sports Personality shortlist, with golfer Rory McIlroy the only one of the 12 nominees not to have won medals in London this summer.
Athletes Mo Farah and David Weir, and sailor Ben Ainslie were also in the mix for the award.
Heptathlete Jessica Ennis, cyclist Sarah Storey, boxer Nicola Adams and swimmer Ellie Simmonds were also in the running.
At what was the biggest night in the event's 59-year history, the first award went to swimmer Josef Craig, 15, who was crowned Young Sports Personality Of The Year.
Martine Wright, who lost her legs in the London bombings on July 7, 2005, and competed in the sitting volleyball at the 2012 Paralympics, won the Helen Rollason Award.
Sports enthusiasts Sue and Jim Houghton won the Unsung Hero Award for transforming a derelict sports ground in Leicestershire into a community facility.
Team GB and ParalympicsGB took Team Of The Year, while Dave Brailsford, the British cycling and Team Sky chief, was the winner of the Coach Award and Usain Bolt was victorious in the Overseas category.