The Rams have now won nine of the last 10 meetings between the sides after Addison rose to head home Gary Teale's right-wing cross in the 89th minute.
It looked like the points would be shared after Alan Judge and Jake Buxton scored their first goals for their new clubs.
Judge, on loan from Blackburn, put Plymouth ahead from the penalty spot in the 17th minute after Buxton had brought him down in front of goal.
But the summer signing from Burton Albion made amends five minutes before the break when he turned home a loose ball from eight yards out.
Derby boss Nigel Clough would have been dismayed to see Steven Davies stretchered off early in the second half.
The former Tranmere forward, who joined the Rams last summer, had his first season at the club disrupted by a catalogue of injuries and was making his first appearance after missing two matches with a hamstring problem.
He was left in some pain with what appeared to be a knee injury following a challenge with two Plymouth players in the 51st minute.
Davies' inclusion was one of two changes made by Clough following the goalless draw at Blackpool in midweek.
Davies replaced Rob Hulse while on-loan Tottenham midfielder Jake Livermore made his first appearance for the Rams after serving a three-match suspension carried over from last season.
Plymouth boss Paul Sturrock also made two changes and handed a debut to a new signing. Karl Duguid came in for Cillian Sheridan after the on-loan Celtic striker suffered a thigh injury in the home defeat to Cardiff.
Kari Arnason, signed from Danish side Aarhus last month, replaced Krisztian Timar in defence.
Duguid went close after 10 minutes when Derby failed to clear their lines but his first-time shot was deflected narrowly wide off the covering Buxton.
Plymouth took the lead when Jamie Mackie burst through and his angled shot struck a post.
Buxton's desperate lunge to thwart Judge prevented the midfielder from simply turning the ball into the net and referee Craig Pawson had no choice but to point to the spot.
Judge picked himself up and smashed his spot-kick into the roof of the net giving Stephen Bywater no chance.
Derby came close to equalising in the 25th minute when Robbie Savage's corner almost curled in at the far post but Rory Fallon headed clear from under his own crossbar.
At the other end, Mackie's shot on the turn had Bywater scrambling across his goal but his effort from the edge of the box went just wide.
Derby drew level from a free-kick on the left after Teale was fouled by David McNamee.
Stephen Pearson's cross was headed goalwards by Davies but Romain Larrieu made a superb save. However, Buxton was on hand to turn the ball over the line from eight yards.
Pearson hooked a right-foot shot just over at the start of the second half before Hulse replaced the injured Davies.
A second period bereft of any clear-cut chances looked like producing a draw before Addison rose to head the winning goal.