The 21-year-old Dutchman has signed a three-year deal with the Seagulls after completing his move for an undisclosed fee.
Ahannach bagged 18 goals last season as Almere made it to the play-offs in the Eerste Divisie before losing to Helmond Sport.
He becomes the seventh addition to Chris Hughton's squad as he looks to defy the odds and keep Brighton in the Premier League following their promotion.
"We are pleased to welcome Soufyan to the club; he's a young player with a lot of talent and good potential," the Brighton boss said.
"He's has done very well playing senior football in Holland in recent seasons and we look forward to seeing more of what Soufyan is capable of over the coming weeks and months."
The confirmation of Ahannach's arrival comes after the club agreed a fee with Club Brugge for Colombia winger Jose Izquierdo while reports also suggest they have made a move for Raphael Dwamena of FC Zurich.