Murray hopes to be back on penalty duty for Brighton
The 34-year-old made it 2-2 six minutes from time after Alexsandar Mitrovic was penalised for handball.
Pascal Gross - who took over spot-kick duties last season after Murray was unsuccessful twice from 12 yards - had a first half penalty saved by Fulham keeper Marcus Bettinelli.
Advertisement
Hide AdAdvertisement
Hide AdGross had already been subbed when the second penalty was awarded and Murray stepped up to send Bettinelli the wrong way.
Asked if he'd like to be Albion's penalty taker again, Murray said: "I would but it's up to the manager.
"It's his call and I appreciate that. We'll see."
On the game, Murray said: "It was crucial to come back. We've put two good performances in against Manchester United and Liverpool. To finish this afternoon on a positive note is good to take into the international break."