Giampaolo Pazzini scored the only goal of the game seven minutes into the second period to close even further a gap that once stood at 13 points.
And it now means that Inter could go top for the first time this season if they beat their bitter rivals when Serie A resumes after next week's international break.
It has been a remarkable comeback for Inter as they search for a sixth scudetto crown in a row having appeared out of the title running in December.
And, following Milan's shock 1-0 defeat at Palermo on Saturday, Italy could even have a three-way fight to the end if Napoli beat Cagliari in Sunday's late game.
That would see the top three separated by just three points.
Inter were far from their best against Lecce as they struggled to raise their game following their dramatic late efforts in the 3-2 success at Bayern Munich in midweek that saw them through to the Champions League quarter-finals on away goals.
Daniele Corvia had the best early chance for Lecce but flicked his shot wide of the near post.
Wesley Sneijder bent a free-kick over the wall at the other end but he too curled it past the near post.
Pazzini struck just a minute after Lecce goalkeeper Antonio Rosati had made a brilliant one-handed save low to his right from the striker's free header from eight yards.
But with his second chance he controlled the ball with his chest as he burst past his marker and fired across Rosati into the far corner.
Julio Cesar had to come to Inter's rescue late on as he blocked a point-blank shot from Andrea Bertolacci.
Meanwhile, Udinese went back above Lazio into the Champions League qualification positions with a 2-0 win over Catania while Juventus won for only the fourth time in 14 league games with a 2-1 success against second-bottom Brescia.
Inter close gap on Milan
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