This is possible using the following process. Here's how you do it:
1) Access the Table Settings on your Submissions page and move the fields you'd like to export over to the 'Visible Columns' column. Move all other fields to the 'Hidden Columns' column. Click 'Save'.
2) Click the 'Share' button and choose 'Tables & Charts'. Click 'Done'
3) Click 'Share Link' and open the link in a new browser tab or window.
4) In the new window, click 'Export All' and choose the appropriate file type.
This will export only the visible columns you chose.
More information on sharing reports can be found here.