The victory lifted newly-promoted Derby off the bottom of the table for the first time since the middle of August with four points from six matches, sending Bolton Wanderers into last place instead.
The winning goal came in the 38th minute when Miller, making a belated debut after being injured following his close-season move from Celtic, volleyed in from outside the box after a long clearance from goalkeeper Stephen Bywater.
Both teams had chances to score with defender David Rozenhal wasting Newcastle's best chance in the first half.
Stephen Pearson had two good chances to give Derby some breathing space after the break but failed to take them.
Victory would have taken Newcastle up to fourth in the table but they dropped a place to 10th after a strangely tepid display.
Michael Owen, who scored three times for England in European qualifiers against Israel and Russia earlier this month, had a quiet game and was substituted after 55 minutes.
The result had no impact on the top of the table with Arsenal (13 points) leading the standings from Manchester City (12), Liverpool (11) and Manchester United (11).