This was the third encounter between the sides this season and for Fury, their toughest after two relatively easy previous victories over WLW.
While the Sony sponsored side shot poorly, it was full credit to a Wolves side who upped their defence to such a level that they led at the half, with a low scoring 31-29 scoreline.
However a 23-13 third period to Fury saw them take an eight-point lead into the final quarter.
After the half, Fury had been sparked by back to back threes from captain Keith Bunyan and the consistent Eddy Leginus.
Fury upped their defence in the final period as they held the home team to seven points, while putting 21 points on the board with Bunyan, Leginus and Connor Martin adding to the team’s three-point tally.
Leginus top scored for the Falkirk side with 22 points, with US star Trey Whitley on 17 and Bantu Burroughs on 14.
Fury head coach John Bunyan said: "Wolves made us work hard for the win and full credit to their players for making this such a close game.
"We got the stops that mattered in the fourth and along with some strong play at that time from us on the offensive end, we were able to pull clear and move to the final eight.”
This weekend will see Fury back to league business with their toughest match to date, at St Mirren, in what will be a battle of the only two remaining unbeaten sides in the Championship.
Meanwhile, ex-Fury star Jonny Bunyan produced a stunning 24-point performance for Glasgow Rocks as they beat Plymouth City Patriots at the Emirates Arena on Sunday.
His former Fury team-mates Ali Fraser and Fraser Malcolm also play key roles as the Fury trio played out the game along with US stars JC Hillsman and Jordan Harris.
Ali Fraser himself continued his dominant form with a 23-point performance in the Rocks’ other win at the weekend when they defeated Bristol Flyers 99-78.