Jerzy Brzeczek's Poland looked as though it was going to finish the day frustrated, but eventually scored the first goal 16 minutes from time in the Group G fixture.
Przemyslaw Frankowski got on the end of Robert Lewandowski's shot and prodded home to send Poland on its way to victory in Warsaw.
Lewandowski was involved again a few moments later, nudging the ball to Arkadiusz Milik while appearing to be fouled and the Napoli man made the most of the advantage.
Euro 2020 will be Poland's fourth successive appearance at the tournament.
They join Italy, Belgium and Russia in wrapping up qualification.