Jurgen Klopp's team caught the hosts off guard with a brilliant start, which resulted in goalkeeper Nick Pope receiving a red card in the 22nd minute, leaving Newcastle with ten men.
This was Newcastle's second loss in the league, the previous one also being against Liverpool at the end of August. Nunez scored in the 10th minute after controlling a long pass from Trent Alexander-Arnold, followed by Gakpo doubling Liverpool's lead seven minutes later from a clever pass by Mohamed Salah.
Pope was sent off after being caught out of his goal by a long clearance from Alisson and grabbing the ball with his hand well outside of his penalty area.
Despite Newcastle's efforts, they were unable to recover, with Liverpool securing their second consecutive league win after a four-match winless streak, bringing them up to eighth place in the table with 35 points, only six points behind Newcastle with a game in hand.
Newcastle's ambitions of making it to the top four are at risk after winning only one of their last seven league matches.