Fifth-placed Fulham had taken the lead at West Brom's training ground when Lucas De La Torre's effort deflected past Alex Palmer just before the half-hour mark.
But the Baggies, buoyed by beating Ipswich Town in the PL Cup last week, scored twice in three minutes late on.
Kyle Howkins equalised with a header on 82 minutes before Alex Bradley struck to take West Brom to within two points of 11th-placed Stoke City
West Brom: 5 Feb v Southampton (A)
Fulham: 2 Feb v Blackburn (H)