The Argentine midfielder answered his contract critics with a stunning winner as Liverpool came from behind at Anfield for the third time in four games this season.
Two costly mistakes from Inter players proved decisive at the Bernabéu, though the Nerazzurri pushed hard for a leveller that never came.
Liverpool host Atletico Madrid at Anfield on Wednesday in the opening match of the UEFA Champions League League Stage.
Turkish Airlines will replace Standard Chartered on Liverpool's kits from the 2027-28 season in a five-year deal the club believes is the most valuable front-of-shirt agreement in Premier League history.
Real Madrid host Inter in the UEFA Champions League League Stage on Tuesday, with both sides yet to register a point in the competition.
The Republic of Ireland forward, handed his Tottenham debut by José Mourinho in 2019, came off the bench to settle a dramatic La Liga contest at the Estadio de La Cartuja.
