The Lowland League champions, 3-0 up after the first leg. went further ahead through Dougie Gair's chipped finish.
Cove levelled on the day through Jamie Watt, but the Highland League champions could make up the remainder deficit.
And Edinburgh will place League Two bottom side Shire over two legs for a place in that division..
Edinburgh City will play East Stirlingshire in the Scottish League Two play-off after completing an aggregate victory over Cove Rangers.