Both sides missed penalties in the nerve-shredding finale but it was Crooks, who had scored for Stanley in extra time, who skied the decisive spot-kick.
The game only really sprung to life in extra time after a goalless 90 minutes, with League Two Stanley more than matching their Championship opposition.
Josh Windass, son of Hull favourite Dean, had a couple of chances in normal time while Crooks headed just over in the 33rd minute.
Hull huffed and puffed without creating a lot, with only Sam Clucas testing keeper Jason Mooney in the 41st minute.
But within two minutes of the restart for extra-time, Arsenal loanee Chuba Akpom, who came on in the 60th minute, cut inside from the right and fired home from 15 yards to put Hull in front.
It was not over, though, as Piero Mingoia burst down the right and his cross was headed home by Crooks in the 105th minute.
Hull replied immediately on 108 minutes when Ahmed Elmohamady’s cross was headed home by another substitute, Greg Luer, but Stanley battled on and on 115 minutes, a strike from Windass was punched out by Eldin Jakupovic and bounced into the net off Terry Gornell.
That sent the match to penalties, with Harry Maguire and David Meyler missing for Hull and Billy Kee and Sean McConville denied by Jakupovic to send the shoot-out to sudden death. Curtis Davies then scored for Hull and Crooks blazed wildly over.
Luer, Tom Huddlestone and Akpom also scored for Hull while Mingoia, Windass, and Gornell matched their efforts.