In an even and keenly contested clash at The Hawthorns, the Baggies enjoyed 57 per cent possession but saw their best efforts thwarted by the excellent Nick Pope.
Just days after being snapped up from Huddersfield, striker Karlan Grant saw a debut goal narrowly chalked off for offside by VAR.
Then after the interval, Pope made a fantastic triple save to deny Grady Diangana, Matheus Pereira and Branislav Ivanovic in quick succession.
Burnley also had their moments, though, with Chris Wood hitting the bar and Sam Johnstone making an excellent stop to keep out an Ashley Barnes header.
And that meant, on the balance of play, a draw was arguably the right result with both teams needing to find a bit more magic in the final third to deserve the win.
Having been hugely disappointed with his team’s display at Southampton, Bilic opted to ring the changes for the visit of Sean Dyche’s side.
In total the boss made six from the trip to St Mary’s with Ahmed Hegazi and Branislav Ivanovic forming a new partnership at the heart of the defence.
Darnell Furlong started at right-back with Conor Gallagher and Filip Krovinovic slightly advanced of a sitting Jake Livermore in a middle three.
Up front, new recruit Karlan Grant came straight into the side with Matheus Pereira on the right and Grady Diangana on the left.
Burnley lined up in a 4-4-2 formation with a strike pairing of Ashley Barnes and Chris Wood.
Former Albion man Jay Rodriguez was named amongst the substitutes.
It was the Baggies who started the game on the front with Bilic’s men a whisker away from opening the scoring in the first few minutes.
A corner routine straight from the training ground saw Diangana find Pereira who then moved the ball onto Krovinovic.
The Croatian then looked to guide the ball into the bottom corner. But from 20-yards, his low effort sailed agonisingly past the post.
While overall the game was fairly even, Albion were dominating possession for the first time this season.
And they threatened again when - making a trademark run through the left channel - Grant collected Pereira’s through-ball.
The former Huddersfield man then charged into the box but under pressure from a defender fired over.
Burnley responded and almost broke the deadlock midway through the half.
Following a free-kick, the ball fell for Barnes who was stood on the corner of the six-yard box.
The forward then lashed a fierce strike at goal that Sam Johnstone did brilliantly to parry away.
Albion wrestled back the initiative with only a narrow VAR call preventing them from taking the lead in the 37th minute.
Diangana’s cross was headed into the net by Grant who, on first glance, did appear to have strayed into an offside position.
And while he had, the review showed it was actually only his boot that was beyond the line with the 23-year-old agonisingly close to a debut goal.
As the two teams made their way off at the break, Bilic’s men had enjoyed 60 per cent possession in what had, overall, been a much-improved display.
But with Burnley dogged and resilient, it still felt as though the clash could go either way with Dwight McNeil particularly bright for the visitors.
The second half quickly settled into a similar pattern as the first with the visitors physicality and resilience giving the impression it would take a moment of magic or a mistake to break the deadlock.
And when the magic did arrive, it was from England goalkeeper Pope who produced a fantastic triple save to keep Bilic’s men at bay.
After Diangana did brilliantly to trick his way past his man at the byline, the shot-stopper produced an incredible one-handed stop to parry away his curling strike.
Pereira then had a go from 20-yards which the keeper again parried - with Pope then making another block as Ivanovic looked to stab the ball in from close range.
A game that had been cagey suddenly then sprang to life with Burnley having two glorious chances to open the scoring.
First, Ashley Westwood whipped in a cross that Wood headed against the crossbar.
McNeil then sent in a dangerous delivery which Barnes met only to see his header spectacularly parried away by Johnstone.
Bilic made his first change in the 72nd minute with Callum Robinson replacing Grant.
Albion were then denied by Pope once again when Townsend sent in a deep cross that was acrobatically met at the back post by Pereira.
The Brazilian did well to steer a volley on target - but Burnley’s keeper was again equal to it and made another smart stop to parry the ball away.
With 10 minutes to play, both teams pressed for a winner.
It was Burnley who came closest to getting one, though, when Johnstone flapped at a corner with the Baggies having to then scramble the ball away.
It meant, when the final whistle blew, a draw was a fair result with the clean sheet a big positive for Bilic’s side.
They will, however, have been disappointed not to have created more in the final third and beat a Burnley team who left The Hawthorns having gained their first point of the season.
Albion (4-3-3): Johnstone, Furlong, Hegazi, Ivanovic, Townsend, Livermore, Krovinovic (Sawyers 79), Gallagher, Pereira (Phillips 88), Diangana, Grant (Robinson 72).
Subs not used: Button, Bartley, Ajayi, O'Shea.
Burnley: Pope; Pieters, Tarkowski, Long, Taylor; Westwood, Brownhill, Gudmundsson (Brady 69), O’Neill; Barnes (Rodriguez 77), Wood
Subs not used: Norris, Brady, Stephens, Vydra, Dunne, Glennon.