There is no way to make this an automatic feature, but you can easily do it manually.
Create a discount coupon that can only be use once by a customer, and only by ONE customer (this is in the settings sections of create a coupon).
Allocate a code to this coupon - make it at least 6 digits/characters long.
E-mail the coupon code to the referring customer when there is evidence that their referral resulted in a new customer signing up.
Be quite clear in your e-mail message that if they allow the code to be accessed by someone else, and it is used, they may forfeit their opportunity to enjoy a discount.
You can put a time limit on its usage, or a minimum value to it as well.
You will need to create a coupon for EACH new customer.
If you want a coupon that can be applied to all referring customers, then just don't limit the coupon to ONE customer (still retain its SINGLE usage per customer though).
This risk here is that if other customers get access to the code, they can use it too - even if they have not referred anyone.