Last year's losing finalists avoided a fifth-round upset at the Broadfield Stadium on Sunday, thanks to goals from Jon Walters and Peter Crouch.
Stoke will now feature in the quarter-finals for the third year running, despite Delap receiving a straight red card for a challenge on David Hunt.
Potters boss Pulis was pleased with the way his side rallied after the 17th-minute sending-off but bemused by the decision.
"You can see why they have been successful," he said of Crawley. "They played to their strengths and were a very, very tough side.
"We had to show them the utmost respect, which we did in our preparation. We are very, very pleased to get past them.
"It is a great achievement by the football club to have made a third successive quarter-final appearance.
"We're one win away from Wembley again and it would be lovely if we could get a home tie.
"We have had three away games in the FA Cup this year so fingers crossed we will get a little bit of luck and get someone at home. I don't mind who."
Asked about the sending off, he added: "There is a big problem within the game in that I think professional people and referees are so far apart in what they see in terms of what are fair challenges or competitive challenges and the bad, reckless challenges.
"We had a fourth official [Lee Probert] today who I think has got Rory sent off.
"He has refereed a game against Liverpool and not even booked a player for what was the worst challenge in the Premier League this year in [Yohan] Cabaye's challenge on [Jay] Spearing.
"Lee was five yards away and the difference between what we think and sometimes what referees think is enormous.
"As a profession we have to try and get closer to the referees to explain and express our views.
"But if you want to go and have a look at Cabaye's challenge and then Rory's today, you tell me which is the worst challenge.
"Like I say, that is the disappointing thing. The sometimes enormous gap between what we think and they think."
Pulis will decide over the next 24 hours whether to appeal Delap's red card, which he reiterated was due to fourth official Probert.
"The referee, I think, gave a throw-in so Lee Probert is the one [that got Delap sent off]," he said. "Lee has told the lads, told my staff, that he is the one but I don't want to be slaughtering people.
"I just want to say there is a discrepancy in what we think and that can't be shown in a greater light than today."
Crawley counterpart Steve Evans also felt aggrieved by today's officiating - in particular referee Mike Jones - after seeing his team exit in the fifth round for the second successive season.
The Red Devils boss believes the penalty from which Walters opened the scoring was wrongly awarded, while he also felt his side should have been awarded two spot-kicks.
"It is fantastic for the town, it is another fantastic run and payday for the football club," Evans said.
"I think we're unfortunate to go out this year. I think I probably take it harder this year than I did going out at Manchester United.
"That is because I have looked at the footage and highlights of the key incidents that have changed this game today.
"I can't help but be disappointed by the match referee. I think from a second-half perspective, he has had a poor game for me and he is normally a very good referee.
"I can only try and second guess as to whether the sending off, which could have gone either way [affected him].
"The referee has perhaps been guilty of allowing that to maybe sway some of his decisions. That's not like him. He is normally top drawer."
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