Daehli turned and fired an injury-time equaliser past Baggies keeper Ben Foster just moments after Thievy Bifouma looked certain to have grabbed a first home win for boss Pepe Mel at the other end.
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.
An outrageous second-minute opener from Morgan Amalfitano and a second from Matej Vydra had threatened to put the home side out of sight as the day looked like it was turning into a nightmare for Cardiff manager Ole Gunnar Solskjaer.
But a long-range effort from Jordan Mutch gave Cardiff a glimmer of hope and Steven Caulker's 73rd minute back-header from Gary Medel's free-kick looked like leaving Mel waiting longer for that first home win.
Thievy had other ideas, turning in Stephane Sessegnon's deflected shot in injury time to spark wild celebrations on the home bench.
But with referee Michael Oliver looking at his watch, there was still time for Wilfried Zaha to make inroads down the right and send in a cross from which Daehli grabbed another 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.
But it was a particularly crushing setback for the Baggies who 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 torrid 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 another superb 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 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 1 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, 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 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.