After thirty minutes, Cody Gakpo gave Liverpool the lead with a short rebound from his own shot, which deflected and diverted Harvey Elliot’s volley from the edge of the area.
However, Justin Kluivert’s header from Alex Scott’s whipped-in corner kick gave the home team a reply on minute 64, ending Jurgen Klopp’s team’s half-time lead.
After replacing Luis Suarez in the second half, Nunez gave Liverpool a 10-minute lead back when he ran down a long ball over the Cherries’ defense, squared up his defender, then cut to his right and beat Andrei Radu with a strong, curving shot.
Nunez, a striker for Liverpool, scored his third goal in as many games for Uruguay.
Liverpool advances to the quarterfinals with the victory after losing to eventual finalists Manchester City in the fourth round of the Carabao Cup the previous season.