The 26-year-old Wimbledon champion beat rugby union player Leigh Halfpenny and jockey AP McCoy to the award.
Murray is training in Miami to prepare for 2014's first Grand Slam, which starts in Melbourne on 13 January.
"I maybe let people down over the years but they stuck by me," he said. "I'll try to repay their faith."
Murray ended a 77-year wait for a British men's singles champion at Wimbledon with victory over Novak Djokovic in July, an achievement recognised as he won Sports Personality with 56% of the vote.
Having finished third in 2012 behind cyclist Bradley Wiggins and heptathlete Jessica Ennis, he became the fourth tennis player to win the award in its 60-year history, after Ann Jones in 1969, Virginia Wade in 1977 and Greg Rusedski in 1997.
"I know I've not always been the easiest person to support but I've been under a lot of pressure for a long time so I'm just really pleased to break through," added the world number four, who won the US Open andOlympic titles in 2012.
"The support I got at Wimbledon this year was by far the best I'd ever had. None of the other Wimbledons could compare to this."
An emotional Murray said he would not immediately celebrate his victory, but would instead visit the gym for a weights session.
He has not played since having back surgery after beating Croatia's Ivan Dodig in September to confirm Great Britain's return to the Davis Cup World Group.
The injury had forced him to miss his first Grand Slam since 2007 when he pulled out of the French Open in May.
"The back is getting much better," Murray told BBC Sport. "It's hard to know how it will respond in a match but the signs have been good and the 10 to 15 days I put in before a match will be key as I'll be testing my body in lots of practice matches.
"As an athlete you're used to throwing your body around and doing physical training, but when you have surgery you need to take a step back and the whole process is tedious and long, but I'm close to coming out the other side of it."
Murray intends to make his comeback in an exhibition event in Abu Dhabi on 26 December before playing in the Qatar Open in Doha ahead of the Australian Open.
His defeat by Djokovic in this year's final in Melbourne meant he became only the second man in the Open era, after Stefan Edberg, to finish runner-up in the event three times.
The other winners were:
- Coach and Team of the Year: Warren Gatland and the British and Irish Lions, who won a series for the first time in 16 years, 2-1 in Australia.
- Overseas Sports Personality of the Year: Sebastian Vettel,who won a fourth consecutive Formula One title in October.
- Young Sports Personality of the Year: Amber Hill, who at 15 became the youngest winner of a senior World Cup in skeet shooting and finished the season ranked number one senior in Great Britain and number five in the world.
- Sports Personality of the Year Diamond Award: Sir Alex Ferguson, who retired in May after more than 26 years as Manchester United manager during which he won 38 trophies.
- Helen Rollason Award: Anne Williams, the Hillsborough justice campaigner, who died in April.
- Sports Unsung Hero: Joe and Maggie Forber for their work promoting basketball in Manchester.
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