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Victory over Belgium in November's final saw Britain win the Davis Cup for the first time since 1936.
Captain Leon Smith guided Britain from the third tier to glory in five years.
British number one Andy Murray played a key role, alongside his brother Jamie, Kyle Edmund, James Ward, Dom Inglot and Dan Evans.
Andy Murray, 28, won 11 of 12 points during Britain's successful campaign.
The award was presented by former pentathlete and 1972 Sports Personality of the Year winner Dame Mary Peters, and Republic of Ireland boss Martin O'Neill.
Captain Leon Smith said: "It is hugely humbling, it is very prestigious. It has been 79 years since we last won the Davis Cup, let's hope it doesn't take another 79 years.
"I hope what it has achieved is we get a lasting legacy for tennis."
Murray joins American John McEnroe and Swede Mats Wilander as the only men to have an 8-0 singles record in a Davis Cup year. In the doubles he is only the fourth man to win 11 rubbers in a single year.
Great Britain's triumphant Davis Cup side won the Team of the Year award at the BBC Sports Personality of the Year show.