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"The team believed (in a win) until the end and this is the most important thing of the evening," Simone Inzaghi told Sky Sport Italia.
Romelu Lukaku started his second stint at Inter with a bang by scoring a brilliant header two minutes into the match.
The Belgian striker, who was instrumental in Inter's title-winning 2020-21 campaign, returned to the club on a season-long loan from Chelsea less than a year after becoming the Premier League club's record signing.
While Inter dominated possession, they spurned a host of chances throughout the game.
"We closed the game with all of our forwards on the pitch, the guys did well in believing (in a win) but the win should have been brought home sooner," Inzaghi added.
Midfielder Barella tried to find Lukaku in the middle of the box with a long-range pass in the 36th minute but Lecce midfielder Alexis Blin rushed out to stop him.
Lautaro, the club's top scorer last season, had the chance to head home a clean cross from Alessandro Bastoni in the 70th minute but the header went just wide of the left post, to the Argentina international's disappointment.