Who mention the salary first, who is the looser.

This is the golden rule regarding salary negotiation. The mistake you made is your give out the salary number too early (how do you know they will not pay higher for this position? how do you know your salary expactition is too high for this position?).
So, better remember, for future interview, don't mention salary untill the last stage, try to potstpone to answer this question till you fully understand the whole benefit package.
Now, the good thing is you are the front runner, they still like you. As I said many times, the real negotiation should start after yiu get the offer letter. make a oppointment with them, go to their office, talk one by one of what you really want, including salary, stock option, sign in bonus, vacation, training, work from home...
If there is no space to go for the offerred salary, how about give me a rise to such such number after 3 (or 6 ) month if I proved I do the excellent job?
Usually only few company give the first offer as the best offer, majority of the company leave a negotiation space in the first offer, anticipate you come back and ask more.
After the successful negotiation, you will feel that this is the most efficient work you have ever done in your life: earn extra $3000 or $5000 or $10000 in just 10 to 15 or 30 minute of talk.
Finally, better put your plan in this way:
1. Top goal: try to megotiate more salary;
2. middle goal: keep the current salary, try to get more in other benefit (if possible).
3. lose the offer

Aiming #1, will be happy if get #2, try to avoid #3.
The employer and HR will respect you more and give what you want to make you really happy if you can show them you are a good negotiator.

Let us know how you did. Good luck.