The Reds' four goals were scored by Dominik Szoboszlai, Jeremie Frimpong, Florian Wirtz, and Hugo Ekitike.
Arne Slot's side will face fellow Premier League side Brighton, the FA Cup website said .
Barnsley had a surprise early on. Just a minute into the match, Davis Keillor-Dunn's header almost put the visitors ahead, but the ball hit the Liverpool crossbar.
Barnsley's pressure continued when Joe Gomez lost concentration, although Nathanael Ogbeta's effort went wide.
Liverpool began to find their rhythm after nine minutes. Szoboszlai made the difference with a powerful long-range shot that beat the Barnsley goalkeeper.
