The matches, between Dundee and either Raith Rovers or Dunfermline Athletic, kick-off at 19:45 BST on 12 and 15 May and will be on BBC Scotland.
The winners will play the 11th-placed finishers in the top-flight, with highlights of both legs of those ties also on the BBC on 19 and 22 May.
BBC Alba will show the Championship and League 1 play-off finals live, too.
The dates for those matches are still to be confirmed.
You can also follow all the action as it happens on Sportsound and on the BBC Sport website and app.
The agreement is part of a new five-year deal in which BBC Scotland can show the Premiership play-off quarter and semi-finals.