Last year's losing finalists avoided an upset against the lowest-ranked side left in the competition thanks to goals from Jon Walters and Peter Crouch on Sunday.
Crawley went out to Manchester United at the same stage last season but almost took a surprise lead when England international Crouch hit his own crossbar.
The Potters' afternoon was made even more uncomfortable in the 17th minute when Delap was sent off for a challenge on David Hunt.
However, the npower League Two side were unable to make the most of their man advantage as Walters slotted home from the spot as half-time approached.
Crouch doubled Stoke's advantage seven minutes after the restart as Tony Pulis' men secured a place in the quarter-finals for the third year running.
Crawley, who previously dispatched of Championship duo Bristol City and Hull, came flying out of the traps at Broadfield Stadium and showed their attacking intent from the offset.
Sanchez Watt and Tyrone Barnett were causing problems during the opening exchanges, with Stoke's defence reduced to some last-ditch challenges to thwart the hosts.
Having seen Pablo Mills direct a header over, Crawley came agonisingly close to breaking the deadlock in the eighth minute.
Hunt's corner caused havoc in the box and a defensive mishap saw the ball rebound off Crouch onto the underside of the crossbar, before Glenn Whelan hacked clear.
The chance seemed to act as a wake-up call for the Potters, who soon began to exert pressure of their own.
Crouch headed clean over the bar in the 10th minute, before Walters flashed a first-time shot across the face of goal moments later.
However, the game changed in the 17th minute when Delap received a straight red card for a challenge on Hunt.
The Stoke midfielder slid in on the right-back around the halfway line and referee Michael Jones deemed it worthy of a sending off.
The visitors' protests did nothing to sway the official's decision as Hunt lay on the deck receiving several minutes of treatment.
The game became scrappy when play resumed as Stoke attempted to adjust to their man deficit and Crawley looked to capitalise.
Kyle McFadzean came close as he forced Asmir Begovic into action with a low 30-yard drive, although the Potters took the ball up the other end and Cameron Jerome blazed wide inside the box.
The match was flowing from one end to the other with both sides missing chances, before Stoke managed to break the deadlock from the spot.
McFadzean upended Ryan Shawcross in the box, allowing Walters to fire home a 42nd-minute penalty - the first goal Crawley have conceded in this season's FA Cup.
Ryan Shotton was booked in stoppage time for a foul on Sergio Torres and was almost joined in the book by Jerome.
Neither side made any changes at the break, although Crawley were soon forced to bring on Scott Neilson for Hunt after the defender seemed to pick up knock.
The hosts were still pressing but conceded a second after Claude Davis brought down Jerome unfairly on the left flank.
The Jamaican received a booking for the challenge and then saw the resulting free-kick by Whelan headed home neatly by Crouch in the 52nd minute.
The second goal dampened the atmosphere inside the stadium and, aside from a Jerome booking for kicking the ball away, the match became flat.
Things livened up again in the 67th minute when Crawley returned to the attack and a goalbound Barnett effort was deflected behind, before a low Dean Howell free-kick was saved by Begovic.
However, Crawley's attempts to get back into the game were leaving them exposed at the back and a Walters effort almost snuck in after it was parried by debutant goalkeeper Rene Gilmartin onto Crouch, before the goalkeeper then palmed it away.
Watt twice came close at the other end, before Stoke returned to the attack and Jerome found the back of the net - only for the goal to be chalked off for offside.
Crawley continued to press forward undeterred by the scoreline as the clock wound down, with McFadzean and Watt both having shots before a deflected Neilson cross hit the top of the bar.
Steve Evans' side camped out in the Stoke half for the final 10 minutes, although they were unable to find a breakthrough as Barnett had an acrobatic stoppage-time effort cleared off the line.