The city council initially told the Balfours to remove the wooden structures, as well as a new timber fence, from the property in Caiystane Gardens, Oxgangs.
However, the family successfully appealed to the Scottish Government and had the decision overturned.
Planning reporter Philip Hutchinson said the facilities were barely visible from the street, largely because of high hedges in the area.
He did, however, urge the homeowners to improve the appearance of the cycle shed and bin unit, and exchange their positions.
He said: “On my visit I found the fence and timber structures to be much less visually jarring than I had initially expected.
“This is in no small part on the account of the mature frontage hedges.
“From the front, the fence is only glimpsed in passing alongside the private footpath which runs back from the street.
“It is clear that the council initially sought to negotiate a compromise. I allow the appeal.”