Not all apartment rentals involve lots of red tape. Many do have a minimum of 5 or more nights, but some are willing to waive that for last minute (definition of which varies from next week to next month or so) and some may require additional cleaning fee for short stays.
If you are in the US a particularly easy agency to use is
Vacation in Paris. Based in US, pay in dollars via cc and get the keys before you leave, staff on the ground in Paris if needed. Have used them and was quite pleased.
Do avoid Craigslist, which is full of scams. Some folks report good luck with airbnb.com but never used myself so cannot offer informed comment.