The switch is in response to the new restrictions imposed by the Scottish Government limiting spectator numbers at outdoor sporting events to 500.
The fixture on Sunday, January 2 was initially due to be staged at the 67,000-capacity national stadium to allow more supporters of both clubs to access a traditionally well-attended inter-city derby match.
However, the rule change which will effectively see the game being played behind closed doors has led to the decision being taken to move it to Edinburgh's recently-opened new stadium in the shadow of Murrayfield, which holds a more modest 7,800 people.
Edinburgh are unbeaten at DAM Health Stadium so far this season, recording three victories and a draw in four United Rugby Championship fixtures.
The first leg takes place at Scotstoun this Monday. The return leg in Edinburgh will be the first between the teams to be held away from BT Murrayfield or Scotstoun since January 2012, when Glasgow played their home game at Firhill.