It looked as though Steve Bruce's men faced more misery after Thursday's deadline day disappointment as Jay Rodriguez's first-half penalty had the visiting Clarets ahead at The Hawthorns until the former Salford forward made his mark.
He lifted into the top corner via a scramble from John Swift's stunning cross for a moment to remember as the roof came off The Hawthorns. The priceless equaliser meant another Championship draw for Bruce's Baggies - but this one felt like a winner against promotion contenders so deep into stoppage time.
The contest was Albion's first and followed less than 24 hours after a disastrous end to the summer transfer window in which the club were unable to get loan deals for midfield duo Josh Onomah and Steven Alzate over the line.
Much was made of the failure to bring in another striker on deadline, but Thomas-Asante, 23, showed the promise of a new blue and white hero with his telling touch.
The headline Albion team news was the omission of Darnell Furlong, who dropped out of the whole matchday squad having been required as an early substitute at Wigan.
It was revealed ahead of kick-off that the full-back's wife had gone into labour, hence his absence.
In for him was Taylor Gardner-Hickman for his first Championship start of the new campaign.
Thomas-Asante, the recruit from League Two Salford on Wednesday, was included on the bench, alongside Kenneth Zohore and youngster Zac Ashworth.
Vincent Kompany's visitors arrived on the back of three wins on the spin in all competitions, including impressive league victories over Wigan and Millwall which had lifted them to third. They included keeper Arijanet Muric in from the off, despite an injury scare.
Albion fans in the Smethwick End wasted little time after the referee's whistle to sound loud unhappy chants towards the club's principle owner Guochuan Lai. They continued through much of the opening half hour.
On the pitch the hosts made all of the running early on in an impressive start. Jed Wallace fired over before skipper Dara O'Shea headed over the top inside five minutes.
John Swift stung the palms of Muric before the Clarets' Ian Maatsen presented Karlan Grant with a golden chance, but the striker was blocked after shuffling on to his right foot.
Burnley, roared on by a vocal Kompany, responded with Vitinho nodding over and a Rodriguez header at David Button, before Albion's Conor Townsend saw a near-post volley parried away by Muric.
Kyle Bartley sent a looping header on to the crossbar from the resulting corner.
Albion were on top despite lively Southampton loanee Nathan Tella flashing a fierce drive narrowly wide across goal.
But, after Grady Diangana shot over with his weaker right foot, things unravelled for the Baggies and they found themselves down to the game's opening goal once again - with a goal laced with controversy this time.
Keeper Muric's long clearance suddenly found Tella through on goal down the middle after Okay Yokuslu played him offside and Bartley lost his man.
Tella went to round Button and hit the deck leaving referee James Linington to point to the spot. Replays showed the keeper touched the ball with his foot.
Rodriguez made no mistake in hammering in his low penalty as Albion fell behind once more.
Bruce's men showed a decent response as The Hawthorns quietened a touch. A flurry of chances before the break should've seen the hosts level. O'Shea nodded a golden chance from Yokuslu's header across goal straight at a grateful Muric.
Seconds later Wallace's low strike somehow cannoned the inside of the post and stayed out after a vital touch from the Kosovo international keeper.
Tella might have punished Albion further just before the interval, but Button denied the speedster from a tight angle.
Burnley were comfortable in the early stages of the second period. They knocked it around and kept the ball well as encouraged by new boss Kompany.
Albion did work a clear chance after the hour but Wallace scuffed a poor effort at Muric. Burnley sub Darko Churlinov flashed a shot wide across goal.
Bruce introduced new boy Thomas-Asante to the delight of the home crowd and he looked bright up front with Grant.
Townsend and Albion, though, survived a huge scare as the former's weak ball pack was pounced on by Churlinov, who struck the post.
Burnley did their best to eat up time into the final 15 minutes - in which keeper Muric limped off. But Albion forged a big chance as sub Jake Livermore sent narrowly wide on the stretch from a half-cleared corner.
Their time-wasting came back to bite, though, as the new Baggies hero sniffed on a moment to remember right at the death for quite the Baggies introduction.
Albion (4-2-3-1): Button; Gardner-Hickman (Phillips, 79), Bartley (Thomas-Asante, 70), O’Shea, Townsend; Yokuslu, Molumby (Livermore, 79); Wallace, Swift, Diangana; Grant.
Subs not used: Palmer, Ashworth, Reach, Zohore.
Burnley (4-5-1): Muric; Roberts, Harwood-Bellis, Taylor, Maatsen (Gudmundsson, 63); Cork (Bastien, 77), Cullen, Brownhill, Tella (Churlinov, 63) Vitinho; Rodriguez (Barnes, 77).
Subs not used: Peacock-Farrell, Egan-Riley, Benson.
Referee: James Linington