Last year’s losing finalists avoided a fifth-round upset against Crawley thanks to goals from Republic of Ireland international Jon Walters and England striker 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.
“It is a great achievement by the football club to have made a third successive quarter-final appearance.’’
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.
“That is the disappointing thing. The sometimes enormous gap between what we think and they think.”
* On Saturday Republic of Ireland defender Sean St Ledger was on the scoresheet as Championship side Leicester claimed a 2-1 win at Norwich to reach the last eight.
St Ledger’s compatriot Wes Hoolahan equalised from the rebound after his spot-kick had been saved by Leicester goalkeeper Kasper Schmeichel, but David Nugent’s second-half strike won it for the Foxes.
Elsewhere on Saturday Everton booked a last eight home tie against Sunderland by easing to a 2-0 win over Blackpool. Royston Drenthe fired home in the first minute for the Toffees at Goodison Park before Denis Stracqualursi was the quickest to react in a goalmouth scramble to double their lead five minutes later.
Bolton also advanced to the last eight after a 2-0 win over Millwall at the New Den, the goals coming from Arsenal loanee Ryo Miyaichi and David Ngog.
CRAWLEY TOWN: Gilmartin, Howell, Mills, McFadzean, 28 Davis, Torres, Simpson, Hunt (Neilson 48), Bulman, Watt, Barnett.
STOKE: Begovic, Collins, Shawcross, Upson, Shotton, Whelan, Whitehead, Delap, Walters, Crouch, Jerome (Wilkinson 84)
Referee: Mike Jones.