Named in Aliou Cisse's starting XI as captain, the Liverpool wideman played 90 minutes as his country claimed top spot in Group H with three important points.
Mane played a key role in Senegal's opener, teeing up Idrissa Gueye for a shot from distance that deflected in off Thiago Cionek to spark wild scenes of celebrations in the stands.
The African nation's second owed more to a lapse in concentration from Poland, however, Grzegorz Krychowiak playing a blind backpass into the path of Mbiaye Niang, who beat the 'keeper to the ball before slotting into an empty net.
Krychowiak went some way to making amends for his error with four minutes of normal time remaining, heading in from free-kick to setup a grandstand finish.
However, Cisse's men held on to record something of a surprise opening win.