The draw means that both sides remain without a victory this season, with Burnley playing one less game than their opponents.
Nick Pope and Sam Johnstone both remained firm in between the sticks to earn their first clean sheets of the campaign in a contest that lacked the quality to win the game.
Despite impressing in the second half, Burnley earned their first point of the season to avoid a club record of four straight Premier League defeats.
Story of the match
With both sides without a win in their opening games this season, it was no surprise that the contest began slowly.
Slaven Bilic made six changes from the side that lost to Southampton, handing Premier League debuts to Filip Krovinovic, Conor Gallagher, Branislav Ivanovic and Karlan Grant.
Grant, who signed from Huddersfield Town on Friday, had the opening opportunity of the game after 20 minutes when he was played through by Matheus Pereira.
However, his left-footed strike arrowed just over the bar.
Ashley Barnes nearly capitalised on Johnstone's poor punch with a lofted effort. Fortunately for Johnstone, it lacked the power and elevation to beat him.
Barnes came close again moments later when his right-footed volley stung the palms of Johnstone from six yards out.
The Baggies thought they had opened the scoring before half-time when Grant directed his header past Pope, only for the linesman to raise his flag as the striker ran off to celebrate.
In the second half, Burnley began the brighter of the two sides. Dwight McNeil was enjoying the space left behind from Darnell Furlong and nearly delivered a perfect cross for Chris Wood two minutes in, only for Johnstone to gather the ball.
Pope denied the hosts with three saves in quick succession. First denying Diangana, before Pereira and Ivanovic were both stopped by the keeper.
Burnley came close to scoring themselves on two occasions through Wood and Barnes.
The first attempt saw the New Zealand international clip the bar, before Barnes forced Johnstone into quick save after being left unmarked at the back post.
Despite being under the cosh for the majority of the second period, the Baggies nearly broke the deadlock. Pereira was denied by Pope after sneaking in at the back from Conor Townsend's cross.
Dwight McNeil is capable of keeping Burnley up
Following his new contract last week, McNeil was one of the stand-out performers in the game.
Partnered on the left with Charlie Taylor, McNeil was taking advantage of the space left by Furlong and delivered teasing crosses on a number of occasions.
His quick, elegant and skilful play has and will cause trouble for Premier League defenders and despite the negativity from Clarets supporters, securing the services of McNeil is better than a new signing.
West Brom are missing a goalscorer
Despite having a goal ruled out for offside and putting in a positive first-half performance, Grant failed to deliver in the second period and was replaced by Callum Robinson.
The former-Huddersfield striker displayed his qualities in glimpses on the night. Quick, strong and not one to shy away from a challenge.
However, due to the physicality of James Tarkowski and Kevin Long, Grant struggled to find space and win the game for his new side.