Nunez left Benfica for Liverpool in a deal potentially worth up to £85million (€100.5m) last month, but struggled to leave his mark in the opening 4-0 pre-season defeat to Manchester United in Bangkok.
The Uruguay international again failed to find the net in the 2-0 win over Crystal Palace, leading Jurgen Klopp to downplay murmurs of impatience and scrutiny by backing Nunez just two games after his arrival.
And the striker soon showed his qualities at the Red Bull Arena, converting from the penalty spot three minutes after being introduced at half-time, before latching onto a Trent Alexander-Arnold pass to finish coolly for his second.
Nunez completed his treble in the 68th minute, displaying an attacking nous that will have satisfied Klopp after ghosting into space to turn home Harvey Elliot's low cross.
But the 23-year-old was not finished there, adding his fourth in the closing stages by curling home after Fabio Carvalho's offload.
Attacking partner Mohamed Salah provided the other strike in what proved to be a routine friendly victory for Liverpool, which next faces Salzburg on Thursday.