The Reds dominated much of the match and went behind through a goal from Herbie Kane, but Crookes equalised for Jack Lester’s men two minutes from time.
There was still time for the Merseysiders to provide more drama as they got the winner in second-half added-time thanks to Adam Phillips.
The game got off to an even start with both sides having decent spells of possession, without either goalkeeper being tested.
Phillips came close to giving the visitors the lead with a 25-yard drive which Jordan Wright watch whisk past his left-hand post.
Soon after, Ryan Yates’ powerful header from a Toby Edser corner went inches wide as Forest looked purposeful in attack.
But on 33 minutes, Liverpool game themselves the advantage when Brooks Lennon bombed down the right midfield and delivered for Herbie Kane who thumped the ball home off the crossbar.
Moments into the second half, Gerry McDonagh nodded into the path of Elvis Otim whose rifle goal-bound was smartly saved by Shamal George’s legs.
McDonagh then had an opportunity of his own as the ball sat nicely on the edge of the area, but the Irishman’s half-volley narrowly evaded the post.
Forest carried on pressing and top scorer Ben Brereton burst into the box before shooting wide.
Despite their attacking efforts, The Reds continued to be frustrated with the final ball lacking. That is until two minutes from time when substitute Liam Adams’ corner was superbly sent in for Adam Crookes who got enough power on it to put Forest level.
However, there would be more drama in injury-time as Ben Woodburn broke forward and played it across the box for Phillips who placed it past Jordan Wright to send his side into the last eight.
Forest: J.Wright, Gabriel, Iacovitti, Nielsen (Gamblen 24'), Edser (A.Wright 82'), Crookes, Yates, Otim, Hayes (Adams 77'), Brereton, McDonagh
Subs not used: Boyd, McClean
Goal: Crookes 88'
Booking: Yates 36'
George, Alexander-Arnold, Owens, Masterson, Parker, Phillips, Lennon, Kane, Woodburn, Alves (T Gomes 46'), Ejaria (Coyle 67')
Subs not used: Kelleher, Dhanda, Johnston
Goals: Kane 39', Phillips 90'
Bookings: Phillips 48', Coyle 87'
Referee: Richard Wigglesworth