In a pulsating conclusion managers Pepe Mel and Ole Gunnar Solskjaer watched on in disbelief as their sides both scored deep into stoppage time.
Albion went 2-0 up thanks to two goals in 10 minutes from Morgan Amalfitano's outrageous effort and Graham Dorrans' clinical finish.
Cardiff then fought back to restore parity in the contest with Jordon Mutch and skipper Steven Caulker on target.
Then, deep in added time, the hosts thought they had won the game through Thievy Bifouma's close-range finish, but the Bluebirds launched a last-ditch attack and 19-year-old Moller Daehli got on the end of Wilfried Zaha's cross to bury his effort and grab a barely believeable equaliser.
At the end of the day a point was what neither side wanted after 90 minutes of high-octane and often kamikaze action.
The late drama summed up a wildly open game in which the visitors were lucky to still be in with a shout after a torrid first 10 minutes.
The Baggies looked set for a productive afternoon when Amalfitano, back in the side after two games out with a knee injury, spotted Cardiff keeper David Marshall off his line and lobbed home the opener from a tight angle on the right.
It could have got even worse for the visitors in a shaky opening period, Marshall going some way to redeeming himself with a vital touch three minutes later to deny an almost certain second for Matej Vydra.
With less than 10 minutes on the clock, Cardiff's nightmare start was complete as Amalfitano sent in a deflected ball from the right to Vydra, whose simple pass across the box was gleefully turned in by Dorrans.
Solskjaer's men had to wait until the 15th minute for their first chance, when Andrew Taylor's low cross from the left evaded Gareth McAuley and almost fell at the feet of Fraizer Campbell.
But having looked completely shell-shocked for much of the first period, Cardiff gave themselves hope on the half-hour when Aron Gunnarsson found Mutch on the right and the midfielder matched Amalfitano's effort by curling the ball over Ben Foster from 35 yards.
For the first time, the home side's nerves were evident although the excellent Amalfitano continued to dictate matters from the right flank, his good work setting up a chance for Youssouf Mulumbu whose low shot was gathered well by Marshall.
Solskjaer introduced another striker in Daehli at the break but it was West Brom who once again made the best start to the second period, Craig Dawson twisting a shot over the bar then getting in the way of an apparently goal-bound effort from Mulumbu.
And Marshall had to get down bravely to deny Vydra at close range in the 52nd minute before James Morrison blazed the rebound high.
The tiring Amalfitano was replaced by Saido Berahino in the 72nd minute as Mel made plain his intention to try to make the game safe - but moments later a Gary Medel free-kick spelled disaster for the Baggies.
Caulker rose highest to reach the set-piece in a crowded box and, while he may not have known too much about it, his back-header looped over the helpless Foster and into the corner of the net.
It marked an astonishing change in fortunes for Solskjaer's men after their nine-minute nightmare, and it was clear their massed ranks of fans behind Foster's goal believed they could go on and claim all three points.
Substitute Daehli wriggled in the left side of the Baggies box and hit a shot straight at Foster, before the late, late drama - celebrations for Thievy's winner were stunningly cut short by Daehli at the other end.