In a captivating encounter at Turf Moor, the Baggies saw Semi Ajayi dismissed on the half-hour mark when, as the last man, he inadvertently handled the ball to deny a clear goalscoring opportunity.
But despite being a man down, Albion went on to create three glorious opportunities to win the game.
Just yards from goal, Mbaye Diange hit the crossbar when it was easier to score after he brilliantly tricked his way past James Tarkowski.
And from a similar distance, Matheus Pereira failed to tap home a Diagne cross before then seeing a shot headed off the line.
In contrast, Burnley created next to nothing with Sam Johnstone not making a save of note.
“The performance deserved the win,” Allardyce said.
“We didn’t do anything other than try to win it even after going down to 10 men, which I’m really pleased about.
“Everybody knows how important the three points would have been.
“We’re all disappointed we didn’t convert one of those because if we had we would have got three very precious points.
“But other than that, I couldn’t be happier with the players’ performance.
“Things are getting better. We are defending better and creating better chances.
“But the sad thing is we are sitting here with two points from the last two games (Manchester United and Burnley) when it should have been six.
“We have still got a way to go to turn these performances into victories.
“Another game has gone by which means we are getting a little bit more desperate when it comes to getting three points in the next game.”