The Poland midfielder converted a spot-kick in the 18th minute and drilled in his eighth goal of the season three minutes later.
Fellow countryman Robert Lewandowski fired in another in the 88th to lift Dortmund to 36 points.
Dortmund keeper Roman Weidenfeller's stunning double-save from a Tomas Pekhart point-blank header and Markus Feulner's rebound shot in the 16th minute denied the visitors an early lead.
"It was good to have come out of this having won 3-0," said Dortmund coach Jurgen Klopp, whose team are through to the Champions League round of 16.
"If Roman Weidenfeller had not made those two saves then Nuremberg would have taken the lead. We had to stay patient and we did that," he told reporters.
Klopp's move to start Blaszczykowski instead of Kevin Grosskreutz paid off with Blaszczykowski calmly converting the spot-kick, awarded for a foul on Lukasz Piszczek.
He soon struck again, connecting with a well-timed cut-back from Mario Gotze as Dortmund enjoyed almost continuous possession.
Nuremberg offered little resistance in the second half but Dortmund wasted several good chances before Lewandowski was left with too much space in the box and grabbed his 12th goal of the campaign.
Dortmund fans reserved their biggest cheer of the evening for playmaker Nuri Sahin, who has returned after spells at Real Madrid and Liverpool and came on as a substitute for his first home appearance in the 77th minute.
