The visitors were reduced to 10 men after half-an-hour when Semi Ajayi was given his marching orders for a deliberate handball, but the Clarets were unable to make their numerical advantage count.
Instead, it was the Baggies who almost wrapped up maximum points as Sam Allardyce's men went for broke after the break.
Ainsley Maitland-Niles fired over the bar when cutting on to his right foot, Mbaye Diagne hit the bar and Matheus Pereira saw his effort cleared off the line by James Tarkowski.
Here are the RATINGS.
1. Nick Pope 6
Made a fine fingertip save to deny Pereira in the first half, but slightly fortunate to finish with a clean sheet at the end. Pereira fizzed one past the post after the break, Diagne somehow hit the bar, Furlong fluffed his lines and Tarkowski cleared one off the line.
2. Matt Lowton 6.1
Got forward well in support in the first half, but given plenty more to think about when Maitland-Niles moved out wide. The Arsenal man was given the freedom of the flank as the Baggies gambled in the second half. Looked tired late on.
3. Charlie Taylor 6.3
Isolated in the first half when McNeil drifted infield, allowing Furlong, Pereira and Gallagher to shut off that side of the pitch. Caught out of position once or twice after half-time, but whipped in a beautiful cross that was crying out for a finish.
4. Ben Mee 6.5
Strolled through the game when West Brom went direct to Diagne, but kept on his toes in the second half as the Baggies went for broke. The visitors set traps and countered at speed as Burnley's back line stepped up.