Marc Skinner's team had several chances to take the lead in the first half with opportunities for Ella Toone, Elisabeth Terland and Melvine Malard but Selma Panengstuen proved to be the star of Brann, keeping the game at 0-0.
United continued with a bright, attacking style of play in the second half but against the run of play, it was the hosts who took the lead through a set-piece with Ingrid Stenevik heading in Signe Gaupset's delivery.
Toone almost found an equaliser late on but the hosts were inspired by the record-breaking home crowd and dug deep to hold onto their lead.