Dario Vidosic’s side will visit Liverpool on the second weekend of May, with time and date yet to be confirmed.
Albion’s men have a confirmed 3pm Saturday kick-off against Wolves on the same weekend.
Goals from Madison Haley and Caitlin Hayes saw the Seagulls women to a 2-0 success at European champions Arsenal in the quarter-finals.
Club chairman Tony Bloom made the trip to Boreham Wood to see the tie.
His deputy, and club chief executive, Paul Barber, said: “It was a fantastic performance.
“All the players performed superbly well. Great credit for the fans that travelled to Boreham Wood to see the game. We had a lot of noise.
“It was great to Tony Bloom there in person to see that as well, which was really good.
“And obviously the draw, of all the teams that are in the competition, we've got a real opportunity.
“Liverpool will be difficult, playing away will be difficult, but it is a real opportunity to reach an FA Cup final, which would be amazing.
“We obviously want to try and get as many fans there as possible
“We will see what we can do to make sure we've got great support.”
Albion were worthy winners and the result is probably only matched in their history by 2-1 win at Chelsea in the league five years ago.
The hope will be that the cup run raises the team’s profile as work continues to secure a new stadium closer to home.
Barber was present when Arsenal won the Champions League last season and said: “That was a great advert for the women's game.
“But I think also it shows you the level of achievement over the weekend in terms of securing the win and the progress that we're making.
“I think it gives us a platform now on which we can help hopefully build our support for our women's team.
“We feel that as good as Crawley has been – they have been great hosts, they have done everything possible to make us feel as welcome as possible - it's not Brighton.
“It's not our city, it's not our club, it's not our stadium.
“It is a challenge to build an audience in somebody else's stadium.
“We saw that when the men's team had to play in Gillingham. We know how difficult it is.
“So our significant ambition is to ensure we've got a home for our women's team in our city.
“It has taken longer than we had hoped. There's still work to be done.
“We're making progress. We are moving steadily forwards on that project but we're a little way away from giving more detail about it just yet.”
Albion are next in action on Saturday, April 25, when they host Manchester City at the Broadfield Stadium.
They will be back at the Amex at the end of the season when Spurs visit on Saturday, May 16.