The award followed manager Claudio Ranieri winning coach of the year after the Foxes won a shock Premier League title last season.
Leicester, who also won their Champions League group to reach the knockout stages this season, picked up the trophy at the ceremony in Birmingham.
Captain Wes Morgan, speaking to the BBC, said: "It's a huge honour to receive this award, what we achieved last season was absolutely incredible.
"No one thought we could achieve it
We proved it was possible
A huge thank you to everyone behind the scenes, they contribute so much to what we strive for and work towards.
"The gaffer has been incredible
He has galvanised us and turned us into a winning team
We had a fantastic season, it was a real team effort for everyone involved with Leicester."
Leicester were 5,000-1 to lift the trophy at the start of last season and Ranieri had also been one of the favourites to lose his job after replacing Nigel Pearson in July 2015.
The former Chelsea manager said: "It's fantastic to take this award here in England, I can say only thank you again to my players, the chairman, all my staff but without our fans it was impossible to win this title
Thank you to everybody."
Swimmer Michael Phelps won the BBC Sports Personality lifetime achievement award after capping his career with five Olympic golds in Rio.
Paralympic swimmer Ellie Robinson, 15, was also named BBC Young Sports Personality of the Year and Marcellus Baz, a boxing coach from the East Midlands, was the BBC Get Inspired Unsung Hero
The Helen Rollason Award was presented to runner and fundraiser Ben Smith.
American gymnast Simone Biles, who won four gold medals and a bronze at the Rio Olympics, was the overseas winner.
Source : PA