Diogo Jota's 78th-minute strike proved to be the decisive moment in the tense tie at the City Ground.
Roberto Firmino missed a glorious chance to put Jurgen Klopp's men in the first half after being played through by a wayward backpass, while Jota had an effort blocked by Forest goalkeeper Ethan Horvath.
Steve Cooper's men should've taken the lead on 78 minutes when Brennan Johnson's cross found an unmarked Philip Zinckernagel but the Dane put the ball wide from eight yards out.
And Liverpool made the home side pay as Jota broke the deadlock as he latched onto Kostas Tsimikas' cross.
The result keeps alive Liverpool's hopes of winning an unprecedented quadruple, however, being drawn against City makes for a very intimidating April.
This was the first meeting between the sides in 23 years and an enthralling encounter was worth the wait as Forest, looking to reach their first FA Cup semi-final since 1991, came flying out of the blocks fired up by a bouncing and confident City Ground crowd.
Liverpool retained a strong spine with Alisson, Virgil Van Dijk, Fabinho and Jota keeping their places from midweek and in the early stages they needed it.
Van Dijk was, typically, a beacon of calm and, knowing what was coming, the visitors looked to take the sting out of the situation and pick off their opponents in the opening stages.
Only Joe Worrall's superbly-timed tackled denied Jota a good shooting opportunity, while Kostas Tsimikas' swerving volley dipped just over.
But for all their fervour, Forest, who left 97 seats empty in memory of the victims of the 1989 Hillsborough Disaster during an FA Cup semi-final between the two sides, really only had a Ryan Yates shot deflected wide to show for their efforts.
Fabinho dragged a shot wide from an Alex Oxlade-Chamberlain corner, while Roberto Firmino squandered the best opportunity of the half when through one-on-one, his decision to go for a cheeky chipped finish proving the wrong one as Ethan Horvath saved.
In between, Van Dijk showed Forest just how they would have to go up a level if they were to threaten the visitors as Djed Spence, the rampaging right-back who had done so much damage in previous rounds, looked to have space to run into the penalty area only to be comfortably ushered off the ball by the Holland captain.
The half ended with Liverpool, lacking both first-choice full-backs Trent Alexander-Arnold (hamstring) and Andy Robertson (illness) having established control in midfield but not in the final third as deputising right-back Joe Gomez blazed over.
Forest started the second half with renewed vigour and Spence's first real incursion into the area won a corner but nothing more.
Hovath saved comfortably from Jota before Klopp made a quadruple substitution in the 63rd minute, sending on midfielders Thiago Alcantara and Jordan Henderson and forwards Luis Diaz and Takumi Minamino.
With Forest legs tiring, the game appeared to be Liverpool's for the taking, but it was the hosts who should have seized victory 15 minutes from time.
Van Dijk vacated his position in central defence to carry the ball into midfield only to lose possession and the hosts countered with a two-on-one which saw Zinckernagel shoot wide with only Alisson to beat.
He was made to pay three minutes later as Jota struck his second important goal of the week following Tsimikas' cross to the far post.
There were some late scares for the visitors when Alex Mighten went down under a challenge from Alisson, but referee Craig Pawson saw no foul and VAR agreed, and then Cafu fired over in added time as Forest went out beaten but unbowed.