The Nerazzurri knew that only a draw here would be enough to finish in the top eight, but they very quickly had a chance to open the scoring as Marcus Thuram flicked the ball past Mohammed Salisu before he was nudged over by Denis Zakaria for a penalty, with Lautaro stepping up to dispatch the spot kick on four minutes.
There were soon calls for a second home penalty as Denzel Dumfries appeared to be bundled over by Salisu, but those appeals were waved away.
Salisu was living dangerously however, again avoiding the concession of a penalty after Thuram went to ground, but things went from bad to worse for Monaco on 12 minutes as Christian Mawissa was deservedly sent off for bringing down Thuram as the France international looked to break away, with Radoslaw Majecki saving the resulting free-kick by Federico Dimarco.
Inter duly doubled their lead though when Monaco couldn’t clear their lines and Lautaro took full advantage, hammering home from 15 yards.
Monaco head coach Adi Hutter therefore decided to make a defensive substitution as Maghnes Akliouche was withdrawn for Caio Henrique, and although Dimarco then blazed over wastefully, Monaco did at least keep their opponents out for the rest of the half.
The match had slowed to a pretty pedestrian pace which largely continued after the interval, but Inter continued to remind Monaco of their quality, notably when Lautaro’s dinked pass found Benjamin Pavard, who drew an instinctive stop from Majecki.
Pavard needn’t have worried, as Lautaro tucked in his third on 67 minutes, capitalising on Majecki’s parry out to him from Henrikh Mkhitaryan’s drive.
Lautaro could easily have added to his tally, diverting Dumfries’ dangerous cross wide, before making way to a standing ovation as teenage first-team debutant Giacomo De Pieri replaced him, while Monaco ended the encounter with nine men as Jordan Teze was taken off on a stretcher with a muscle injury.
Inter finished the evening fourth in the table, while Monaco disappointingly slipped to 17th, meaning they will be unseeded for the knockout play-off draw, having now won just one of their last 11 UCL main-draw away games.