The world No 2, who was picking up the honour for the second time, led Great Britain to their first Davis Cup title since 1936 last month.
Accepting the award the Scot said: "I didn't expect this - a friend actually sent me a message the other day with an article from a newspaper which said Andy Murray is duller than a weekend in Worthing, which I thought was a bit harsh - on Worthing.
"It's very humbling to be up here in front of so many great athletes - I'm just a great sports fan and I'm really nervous."
"This has been a five-year journey - we were right down at the bottom level or tennis and now we're ranked number one in the world and I never thought that would be possible."
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In second place came Kevin Sinfield, who retired from rugby league after an extraordinary career, including winning the Challenge Cup, League Leaders' Shield and the Super League title in his final season, before switching codes to union
Jessica Ennis-Hill, who was crowned world heptathlon champion in August, 13 months after giving birth, came in third. Jump jockey legend AP McCoy received the Lifetime Achievement Award after bringing his illustrious career to a close in April.
Former All Blacks fly-half Dan Carter picked up the BBC Overseas Sports Personality of the Year award. The 33-year-old Carter ended his international career by leading New Zealand to a World Cup triumph in England in the summer.
Northern Ireland manager Michael O'Neill won the Coach of the Year award after leading his nation to the Euro 2016 finals. And the British Davis Cup team were rewarded for their historic triumph in Belgium by winning the Team of the Year trophy.