The erotically charged thriller rated 15 in the UK, generated 40 complaints from audiences.
According to the BBFC's annual report, "some complainants had expected to see a film about ballet".
Instead, it continued, the Natalie Portman film depicted "the story of a young woman's mental disintegration".
The BBFC announced it had commissioned a major research project - the first for 10 years - into public attitudes over sexual violence in films.
The report is due to be published later this year.
Presenting the annual report in London on Wednesday, BBFC director David Cooke said the 40 complaints for Black Swan constituted a "very small number".
By contrast, 2012 release The Woman in Black has received around 120 objections to its 12A rating.
Portman won the best actress Oscar in 2011 for her portrayal of a ballerina who loses her grip on reality while dancing Swan Lake.
What ruffled the feathers of most complainants, according to the BBFC, was a sex scene between two female characters.
"While the scene is visually discreet, narratively justified and within the 15 guidelines criteria, some correspondents felt it was pornographic in nature," the report reads.
"That it was a sex scene between two women was an aggravating factor for some who argued that portrayals of homosexual activity should either be restricted to the 18 category, or not shown at all.
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