"It was an emphatic victory for Fabulous Frank," said Gambaccini. "It wasn't even close."
Elvis Presley came in second, followed by Nat King Cole, Ella Fitzgerald and Bing Crosby.
Some 44,756 votes were cast by the public, but BBC Radio 2 also consulted a panel of singers and music experts.
Sinatra is not a surprise winner - the man who sang such classics as My Way and Strangers in the Night was adored by millions.
The New Jersey-born son of poor Italian immigrants started out as a singing waiter, but by his twenties he was performing with Bing Crosby and the famous singing Rat Pack.
His cheeky good looks drove the girls wild and landed him film roles in acclaimed films including From Here To Eternity (1953), which won him an Oscar for Best Supporting Actor.
Unsurprisingly his best-known film role combined acting and singing in Guys and Dolls opposite Marlon Brando.
Sinatra's private life was notorious.
He had a tempestuous marriage to actress Ava Gardener and his marriages to Nancy - the mother of his three children - and actress Mia Farrow also broke down.
He finally settled down with Barbara, the widow of Zeppo Marx - but his children contested his bequests to her in his will.
Sinatra spent much of his career reacting to sounds and rhythms that he did not like.
"The most brutal, ugly, degenerate, vicious form of expression it has been my displeasure to hear," Sinatra wrote of rock 'n' roll.
Interestingly the number two Voice of the Century according to the poll was a Rock 'n'Roll hero - Elvis Presley.
Presley's continuing popularity put him at top place in a recent Forbes Magazine list of highest earning dead celebrities.
Another crossing on the top ten list was Presley and the number six - John Lennon.
Lennon met Presley with The Beatles in 1965 and he later wrote that Presley "was only one person in the United States that we really wanted to meet".
The most popular woman on the list was Ella Fitzgerald, who was ranked fourth.
Gold albums
Fitzgerald was known for her clear voice which could glide from low notes to high, from be-bop to ballads. The so-called "first lady of song" died aged 78 in 1996.
Most of the top 10 singers are now dead with two exceptions - both female.
Soul singer Aretha Franklin came in at number seven and she plans to expand an extraordinary career by recording an album of operatic arias later this year.
And the popular singer and actress, Barbra Streisand, came in at number nine.
With 43 gold albums, Streisand is exceeded only by Elvis Presley in the all-time charts.
1: Frank Sinatra
2: Elvis Presley
3: Nat King Cole
4: Ella Fitzgerald
5: Bing Crosby
6: John Lennon (The Beatles and solo)
7: Aretha Franklin
8: Billie Holliday
9: Barbra Streisand
10: Freddie Mercury (Queen and solo)
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Internet links: BBC Radio 2 Voices of the Century | John Lennon on Elvis Presley | Ella Fitzgerald | Nat King Cole | The Sinatra Family | Bing Crosby |
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