Sergio Conceicao's team, competing without the suspended French goalkeeper Mike Maignan, found themselves down 2-0 for more than an hour due to a brace from Lecce's Montenegro striker Nikola Krstovic, who scored at the seventh and 59th minutes.
An own goal by Antonino Gallo in the 68th minute provided a glimmer of hope for the northern side, and five minutes later, Pulisic converted a penalty to level the score.
The US international then secured his team's first victory since February 15 by netting his second goal nine minutes before the final whistle.
The previous season's Serie A runners-up have now ascended to eighth place, yet they remain 14 points adrift of the champions and current leaders, Inter Milan, who are set to face bottom-placed Monza later on Saturday.
Conceicao's future remains uncertain despite the victory, as he took over from fellow Portuguese coach Paulo Fonseca at the end of December.
Reports indicate that the management of Milan is engaged in advanced discussions with former Juventus coach Massimiliano Allegri.
