Norrie survived a scare on Friday in a tight contest against Tomasz Berkieta, ranked 511 places below him, to get Britain off to a winning start.
But with the tie in Britain’s grasp, the world number 34 produced an efficient performance against Pieczkowski.
A break in the ninth game of the first set proved decisive as Norrie held his nerve to serve out the opener to love.
As he did against Fery on Friday, Pieczkowski showed signs of fatigue in set two while Norrie’s level did not waver and he won every point behind his own serve.
The 30-year-old seized his one and only break opportunity in the seventh game and held serve from that point on without any concerns to wrap up a polished victory and put the tie to bed.
That result will have come as a relief to Cash and Glasspool, who had started well in the doubles but ultimately lost out in a dramatic tie-break.
They were perhaps feeling the pressure of representing their country for the first time in Davis Cup, as well as the effects of playing a significant volume of tennis this year – including historic wins at Queen’s and Wimbledon before their recent shock first-round exit from the US Open.
Dan Evans, a member of Britain’s 2015 Davis Cup winning squad, said of the doubles pairing on Tennis Channel: “I have to be honest, I thought the British guys together as a team were pretty poor. They were not as one out there.
“They were not bouncing off each other, not helping each other enough, and the Polish team were a better team.”