Giuseppe Sculli struck twice at Stadio Via del Marea to end Genoa's run of six consecutive away defeats.
The 30-year-old gave the visitors the lead midway through the first half, but second-half Lecce goals from Luis Muriel and Davide Brivio had looked to have consigned Alberto Malesani's men to further away-day misery.
With five minutes remaining Sculli struck again to earn a well deserved point.
Sculli has now scored eight goals this season, four of them coming in Serie A.