Stefano Okaka, a second-half substitute who is on loan at Bari from Roma, notched the game's only goal 11 minutes from time at the Stadio Via del Mare to move the Galetti within a point of their second-bottom hosts.
Lecce's miserable afternoon was compounded in the last minute by the dismissal of Guillermo Giacomazzi for a second bookable offence.
Ruben Olivera had the first sight of goal for the home team after 11 minutes but blasted his 20-yard free-kick just wide.
Three minutes later Lecce goalkeeper Antonio Rosati had to be alert to keep out a powerful header from Vitali Kutuzov.
The hosts continued to push for the opener, but to no avail. First Ignacio Piatti fired over just outside the area and then Olivera lashed his shot wide after a neat one-two with Edward Ofere.
Javier Chevanton also tried his luck four minutes before the break but his effort from the edge of the box did not trouble Jean-Francois Gillet.
The game continued in the same vein after the interval with Lecce mustering the better of the early second-half opportunities.
Gillet kept out a fierce left-footed attempt from Djamel Mesbah and then Brazilian Fabiano headed off target from a Chevanton free-kick and substitute Daniele Corvia shot wide.
Okaka, though, did not take long to make his presence felt after being introduced for Francesco Caputo and soon got himself an opportunity to break the deadlock but could not find a way past Rosati from close range.
From his next opportunity, the visitors went one up. The 21-year-old latched onto Alessandro Gazzi's superb assist and beat Rosati from the left side.
Fellow debutant Kamil Glik also did his bit by clearing an Ofere effort off the line six minutes later.
Giacomazzi was then given his marching orders, picking up a second yellow card for fouling Gillet, as Lecce's afternoon went from bad to worse.