The Baggies saw defender Semi Ajayi dismissed on the half-hour mark when, as the last man, he inadvertently handled the ball to deny a clear goalscoring opportunity.
After that the hosts were expected to go on and win the game.
Instead, Albion dominated the chances created and had three glorious opportunities to win it.
First the impressive Ainsley-Maitland Niles collected the ball in the box before working himself some space before firing over.
Mbaye Diagne then tricked his way past James Tarkowski before powering a strike against the crossbar when it was easier to score.
But while that was an incredible miss - what followed was even more remarkable.
Diagne squared for Matheus Pereira to have a simple tap-in but the ball hit the Brazilian’s heel.
Maitland-Niles kept the move alive and found Pereira again who, this time, did get a shot away just yards from goal.
But goalkeeper Nick Pope somehow got something on the ball and that meant Tarkowski was able to head it clear from the goal-line.
In stark contrast, Burnley created next to nothing over the 90 minutes with Sam Johnstone not having a save to make.
But while Sam Allardyce will be proud of his team’s overall display, he knows Albion need wins and fast - with this a huge opportunity they let slip by.
Allardyce made two changes from the team that had put in the best performance of his tenure to draw with Manchester United last weekend.
In what was a major blow, winger Robert Snodgrass missed the trip to Lancashire with a bruised knee and was replaced in the starting XI by Matt Phillips.
Darnell Furlong also earned a recall at right-back with the 25-year-old offering more of an attacking threat than Lee Peltier who dropped to the bench.
Despite the changes, Albion continued in a 4-3-3 formation with Matheus Pereira on the right, Phillips on the left and Mbaye Diagne the loan centre-forward.
In midfield, Okay Yoksulsu screened the back-four with Conor Gallagher and Ainsley Maitland-Niles more advanced.
Burnley entered the match having been decimated by injuries with key men Johann Berg Gudmundsson, Robbie Brady, Chris Wood and Ashley Barnes all not involved.
They lined up in their customary 4-4-2 system with former Baggies stars Jay Rodriguez and Matej Vydra up front.
In what proved to be a stop-start and scrappy opening 20 minutes, both teams created half chances to open the scoring.
For Albion, Pereira sent a free-kick from 25-yards straight into the arms of Nick Pope.
Furlong also sent a left-foot drive from the edge of the box narrowly past the post.
While for Burnley, Rodriguez managed to squeeze in between Semi Ajayi and Kyle Bartley to meet a Matt Lowton cross.
But around 12-yards from the goal, the forward glanced a header well wide of Sam Johnstone’s post.
Overall there had been a lack of quality from both sides - with the teams not helped by some blustery conditions.
But when Albion did get the ball down and play they looked bright.
And they came close to opening the scoring when Maitland-Niles won the ball high for finding Diagne who cleverly flicked the ball to Pereira.
Around 18-yards out, the Brazilian then arrowed a low drive that was heading right into the bottom corner.
But Pope showed his class by getting down and brilliantly parrying the ball away for a corner.
It felt, at that moment, Allardyce's men were in the ascendancy.
But on the half-hour mark they were reduced to 10-men when Ajayi was somewhat harshly sent off.
The drama began when Gallagher first prevented the ball from going out for a throw-in before then losing possession with a dreadful pass towards Ajayi.
It meant Burnley could break two on one with the Nigerian international having to get something on the ball to prevent Vydra from being clean through on goal.
At first glance, Ajayi looked to have swept up the danger expertly. But VAR called it ball after spotting a hand-ball.
And after Mike Dean checked his pitch-side monitor, he opted to send the former Rotherham man on his way.
Allardyce responded by introducing Dara O’Shea for Phillips.
It meant Albion switched to a 4-4-1 system with Gallagher on the right and Pereira on the left.
And the rest of the half simply petered out with Burnley have struggled to make their man advantage count.
Dyche clearly told his players to play with more width during half-time.
And the hosts saw a big appeal for a penalty waved away when a cross hit the arm of Bartley - VAR however decided there should be no spot-kick.
But while Burnley did look more dangerous, they still hadn't truly tested Johnstone as the clock ticked past the hour mark.
And Albion then came very close to nicking a goal on the break with Pereira cutting in from the flank before firing narrowly wide from 18-yards.
The Baggies were now playing a 4-3-1-1 system with Pereira just behind Diagne.
And within the space of 90 seconds they created two glorious opportunities to open the scoring.
First Pereira found Maitland-Niles in the box who did well to cut inside and work himself a shooting opportunity.
But around 12-yards out the Arsenal loanee fired over when he really should have done better.
Then just moments later a swift counter saw Diagne race into the box and go past James Tarkowski as if he wasn’t there.
The Senegal international then simply had to fire past Pope and into the net but incredibly hit the crossbar when it was easier to score.
You felt at that point Burnley’s goal was living a charmed life but what followed was remarkable
The increasingly influential Maitland-Niles slipped Diagne through one-on-one.
Instead of shooting he tried to find Pereira for a tap-in but the ball hit his heel.
Maitland-Niles managed to recycle it and find Pereira again who was just six-yards out.
And his shot was then headed off the goalline by Tarkowski after Pope had managed to get something on it to slow it down.
From there Albion kept pressing for a winner.
But in the end Burnley did enough to secure a clean sheet with Baggies ultimately heading home having missed a huge opportunity to claim three crucial points.
Burnley (4-4-2): Pope, Lowton, Mee, Tarkowski, Taylor, McNeil, Cork, Westwood, Brownhill, Rodriguez (Mumbongo 81), Vydra.
Subs not used: Peacock-Farrell, Norris, Bardsley, Long, Nartey, Dunne, Benson, Glennon.
Albion (4-3-3): Johnstone, Furlong, Ajayi, Bartley, Townsend, Yokuslu, Gallagher, Maitland-Niles, Pereira, Phillips (O’Shea 32) Diagne.
Subs not used: Button, Peltier, Livermore, Sawyers, Diangana, Robinson, Grant, Robson-Kanu.