The Network Rail scheme includes a glass canopy and four escalators to take passengers from a plaza off Princes Street to Waverley station.
Permission was granted despite concerns over its impact on the world heritage site.
The rail company welcomed the decision, saying it will provide passengers with a "warm and inviting entrance to the capital". Director Ron McAulay added: "Commuters will no longer have to struggle up the windswept Waverley Steps with luggage or pushchairs. Easy access will be available to everyone."
The canopy will be supported by steel "trees" and a footbridge will connect to the existing route through the station. A stone arch at the foot of the steps will be dismantled and rebuilt elsewhere in the station.
City council planning convener Trevor Davies backed the project, arguing that the city should not be set in aspic. "There is a lot that is not right in the centre that we need to change," he insisted.
But the plans were opposed by the owners of the nearby Balmoral Hotel, a grade I listed building. They complained that views from the hotel to Edinburgh Castle would be blocked by the structure.
Local councillor David Guest maintained that it is wrong to consider the scheme in isolation. "It is like designing the front door of a house without designing the actual house, slap bang in the middle of a world heritage site and conservation area," he complained. "It is crazy to go ahead with this without knowing what the station will look like."
The scheme is the first stage in phase one of the station's redevelopment.
Listed building consent is being sought for three platforms to increase capacity and reduce delays. Network Rail hopes that the alterations can be completed by the end of 2007.
McAulay commented: "Waverley is the biggest station in the UK outside London and this redevelopment reflects its importance as a crucial commuter hub and landmark."