I can create my query in SSAS and execute, but I want to export the result to excel or a .csv file so that I can analyse it further.
There doesn't seem to be an intuitive way to do this and PowerPivot is not able to provide what I want.
You can create a pivot table in Excel that is linked directly to your cube. This is shown e. g. in this video. Basically, you need to install the Analysis Services client on your computer (which will already be done in case you have the SSAS developer client). Then you select "Data/Form other sources/From Analysis Services" in Excel, and follow the Wizard to configure the connection to your SSAS cube, and to save this information in a llocal file, and then select the cell, where the pivot table will be located.
Another approach would be to use cube formulas like CUBEMEMBER and CUBEVALUE in Excel cells which allow you to directly get information from a cube to a cell. You can find their documentation somewhere in the Excel help or here. You can even convert a Pivot table to these formulas by selecting "Options/Tools/OLAP Tools/Convert to Formulas" as described here for Excel 2007, but there is no way back.