A dominant Fulham side appeared to be heading towards a first top-flight win since 30 November when Bobby Decordova-Reid opened the scoring in the first half.
But a Kyle Bartley equaliser two minutes after the break shifted the momentum towards the hosts.
Sam Allardyce’s side looked to have engineered a complete turnaround when Matheus Pereira flicked in a cross from new loan signing Mbaye Diagne.
However, Ivan Cavaleiro headed in an inviting cross from fellow substitute Harrison Reed to earn the visitors a point.
The result leaves Fulham 18th in the table, while West Brom – who gained their first point at home since Allardyce took charge in December – are in 19th, two points further back.