My plan always goes by the "date of service" not the day you paid.
So, if the procedure was performed in December 2010, it should be claimed against your 2010 benefits.
Edited to add: This doesn't really apply to your question, but some plans will also let you bill "dates of service" that occurred in Jan-March of one year back to the previous year's flex plan. For instance, if I have a procedured done on February 15th, I can still claim it against my 2010 flex plan if I still have money left over OR I can claim it against 2011. But that's not what you asked....
In your case, it goes to the 2010 plan.