Aleksandar Mitrovic put Fulham ahead twice at Craven Cottage as Marco Silva's team threatened a major upset, but Nunez showed just why Liverpool spent big to bring the Uruguay forward in from Benfica in the summer.
Nunez first met a low cross from Mohamed Salah with a deft backheel flick into the top of the net to make it 1-1 in the 64th minute, and then helped his Egyptian teammate net the second equaliser in the 81st minute.
Liverpool captain Jordan Henderson then came close to an injury-time winner when his curling long-distance shot hit the crossbar.
Mitrovic headed in the opening goal in the first half, rising above Trent Alexander-Arnold at the far post to meet a cross from the right, and then converted a penalty in the 72nd minute after he was brought down by Virgil van Dijk.
Player ratings from Craven Cottage can be seen in the gallery above. To see the next rating, just swipe.