The critically acclaimed drama follows Calum, played by Normal People star Mescal, and his daughter Sophie, played by newcomer Frankie Corio, during a holiday in Turkey.
Celia Rowlson-Hall portrays an adult Sophie, who looks back at the holiday 20 years on.
Director Charlotte Wells won three awards – best director, best debut director and best screenplay.
The award for best British independent film was presented by Daisy Edgar-Jones, Mescal’s co-star in hit BBC series Normal People.
These were in addition to three craft Bifas that Aftersun was previously announced as winning.
Aftersun had led the nominations with 16 nods in the awards’ first year with gender neutral acting categories.
Best lead performance went to Rosy McEwen and best supporting performance to Kerrie Hayes for their roles in Georgia Oakley’s 1980s-set Blue Jean.
The ceremony also saw Oakley awarded best debut screenwriter.
Breakthrough performance went to Safia Oakley-Green for The Origin, Andrew Cumming’s feature about a nomadic tribe facing a terrifying ancient threat.
The award for best joint lead performance went to Tamara Lawrance and Letitia Wright for their roles as siblings who communicate only with each other in The Silent Twins.
Our River…Our Sky, about neighbours caught in Baghdad during the civil war of 2006, scooped the prize for best ensemble.
Elsewhere, best international independent film went to Joachim Trier’s The Worst Person in the World.
The Richard Harris award for outstanding contribution by an actor to British film was presented to Samantha Morton following a 30-year career in acting and directing.
The ceremony was held in Old Billingsgate, London, and hosted by Ben Bailey Smith.
Open Door, which works to help young people without financial support gain places at leading UK drama schools, was presented with the special jury prize by jury member Jenna Coleman.
As part of the organisation’s 25th anniversary, the body has made the move to award five acting accolades with no mention of “actor” or “actress” as well as adding a selection of new performance awards.
Bifa was created in 1998 and has since celebrated and promoted British independent cinema and filmmaking talent in the UK.