The Tangerines secured a deserved place in the fourth round of the competition after the Baggies put in a thoroughly underwhelming display at Bloomfield Road.
Despite their Premier League status, Albion had to battle back twice to force extra time.
Jerry Yates gave Blackpool a first-half lead before Semi Ajayi pulled Allardyce’s side level after the interval.
Gary Madine restored the host’s advantage before Pereira again leveled with a penalty.
But while both teams created chances in extra-time, it was Blackpool who showed more character and grit.
And they won it on penalties despite Yates seeing the very first spot-kick saved by David Button.
Kamil Grosicki scored for Albion but misses from Kyle Edwards and Darnell Furlong put the hosts on the cusp of victory.
Albion got another shot at it when Sullay Kaikai missed what would have been a winning spot-kick.
But Pereira then saw his following effort saved to ensure it was the League One side who made it through.
"The battle was always going to be tough,” Allardce said.
“I saw exactly how tough it was going to be by Blackpool using their assets, playing off big Madine up front, who causes a problem and we have to battle in that area.
"But the most disappointing thing for me was that the goals didn’t come from that area, they came from our lapses in concentration.
“Both their goals were getting through the middle after we gave the ball away.
“We came back twice with goals from Semi, good header, and the penalty.
“Extra time came and both sides went out to try and win it.
“We got the best possible start with Butts’ saving the first penalty.
“Then from our point of view, I’m hugely disappointed in the penalty takers that missed.
“When you save the first penalty and put yourself one-nil up, at the level we’re supposed to be at, we shouldn’t have lost at all. I am disappointed with that.
The effort from the players was fine. I’ve seen players I’ve not seen before. I’ll make a judgment on them in the next few weeks.
“And then we’ll move forward as best we can under these restrictions and we’ll be working as hard as we can to bring new players in.”
Allardyce admitted he expected his players to perform better in Lancashire.
And he also confirmed Charlie Austin is on the brink of leaving the club for QPR.
“The chance was to win it at penalties, it gives you a win and you go home reasonably satisfied,” the boss said when asked if the win would have boosted confidence.
“I say reasonably satisfied because I did expect us to play better than we did.
“I didn’t expect us to have to come from behind twice.
“We hung in there with our effort and commitment. But we lost on penalties in the end and I will make a judgment on the players as quickly as possible now.
“Then we will be off into the market to try and find players to make the squad better.
“We have let Charlie go to QPR. That creates an opening in terms of finances to find a centre-forward and then we will look at other areas when we brought in a centre-foward.”