Not since 2001 had Nantes added to its list of honours, but it will play in the UEFA Europa League next season after edging this showdown at Stade de France.
Blas was on target from the spot in the first minute of the second half after Hicham Boudaoui handled and Stephanie Frappart – the first female to referee a men's Coupe de France final – pointed to the spot.
Nantes held on to to secure a return to Europe, leaving Nice without a trophy since back in 1997.
Amine Gouiri set the tone for a poor first half when his sloppy touch in the penalty area gave Nantes an early let-off.
Blas tried his luck at the other end from just outside the box, but failed to generate enough power on a shot that was saved by Marcin Bulka.
Gouiri's deflected strike flashed wide and Andy Delort should have done better with a header from the resulting corner, with Nantes goalkeeper Alban Lafont flapping.
It was Les Canaris who took the lead right at the start of the second half, Blas emphatically dispatching his spot-kick after Boudaoui handled in the box.
A desperate last-ditch block prevented Moses Simon from doubling Nantes's lead soon after they went in front and Jean-Charles Castelletto headed wide, with Nice looking vulnerable at the back.
Lafont palmed away Delort's drive and showed great reactions to deny Delort following up with 20 minutes to go as Nice failed to conjure up an equaliser.