The son of France World Cup winner Lilian Thuram, who won a series of honours in Italy as a defender with Parma and Juventus, the 26-year-old forward moved to Milan from German club Borussia Moenchengldbach in July.
He met Federico Dimarco's cross with a forceful, diving header after 23 minutes to put the Champions League finalists on their way to a convincing win.
It turned out Thuram was just getting started.
Having been involved in both goals of a 2-0 win at Cagliari on Monday, he went on to set up Lautaro Martinez for Inter's second and then drew the penalty that Hakan Calhanoglu converted for the third.
"I liked the way he played in the other games and I was hoping he would score," Inter coach Simone Inzaghi said of Thuram. "I'm very satisfied."
Martinez added the final flourish with an expert touch to redirect a free kick from Juan Cuadrado and raise his league-leading tally to five goals this season.
Inter joined city rivals AC Milan as the only teams to have won all three of their matches so far - with the Milan derby up next on September 16 following the international break.
It's the first time since 1966 Inter have opened a league campaign with three straight victories and three consecutive clean sheets.
Having finished 16th last season to narrowly avoid relegation, Lecce have begun the new season at a cracking pace and remain unbeaten after three games.
In-form Montenegro striker Nikola Krstovic opened the scoring with a glancing header for the southern club, with Gabriel Strefezza adding the second for Lecce eight minutes into second-half added time.