The Baggies were lucky to survive until the shootout as their determined hosts enjoyed the better of it on a cold day by the sea.
Both sides created chances in busy first 45 minutes with Jerry Yates confidently tucking home before the break.
The visitors did haul themselves back into after the restart as Semi Ayjai nodded home.
However, they were unable to build on that the hosts continued to dominate with cup hotshot Gary Madine slotting them back in front.
Allardyce’s team were given a lifeline in the final minutes as Matheus Pereira tucked home from the spot.
But again they failed to grasp the nettle as Kyle Edwards missed a brilliant chance in extra time before penalty misses from Edwards, Pereira and Darnell Furlong sent Blackpool through.
Full Time at Bloomfield Road it’s Blackpool 2 West Brom 2 with the Seasiders winning it on penalties.