The eccentric film, which features salesmen flogging novelty items, singing bar women and a sex-mad flamenco dancer, was hailed by critics in Venice for its distinctive look and moving exploration of what it means to be human. The film features a series of comedic sketches exploring the human condition.
Critics had widely tipped Venice's top prize for Joshua Oppenheimer's powerful "The Look of Silence" documentary on the Indonesian genocide, which missed out but instead took home the prestigious Grand Jury prize.
Jury head Alexandre Desplat, a French film score composer whose works include the music for "Harry Potter and the Deathly Hallows", said "Look of Silence" and "A Pigeon" were "two sides of the same coin. Both works had a an incredibly powerful impact."
Sweden's Andersson told the audience in English that he had been inspired by Vittorio De Sica's 1948 "Bicycle Thieves" — an emblem of Neorealism filmmaking — particularly the scene in the pawnshop, because the character "discovers Rome's poor".
"It's a humanistic scene, it's true empathy. That's what a movie should be," he said.
The work is the final part of a trilogy including "Songs from the Second Floor" and "You, the Living" and has a theatrical feel, with touches of Monty Python-esque comedy.
Props are limited to the odd briefcase in surreal scenes which jump from "Limping Lotta's bar" to a cafe where Sweden's 17th-century militarist King Charles XII seduces a waiter.
"It's the first time a Swedish film wins the Golden Lion, I'm hugely proud," Andersson said.