It was a game of two halves in the Barclays Women's Super League, as United dominated the first while Brighton piled on the pressure in the second.
Skinner's side took the lead early on through Grace Clinton, who turned in Celin Bizet's cross just 10 minutes into the tie.
This effort cancelled out by former Red Nikita Parris, who departed in the summer, as she headed in an equaliser shortly after half-time.
FIRST HALF
Brighton dominated possession in the opening five minutes but it was the Seagulls keeper and former Red Sophie Baggaley who was forced to make the first save.