From the facts you've given, it appears that you can "fix" your husband's papers without him having to leave the US. Very basically, here is the process. First, you file an I-130 family petition. You are the petitioner and he is the beneficiary. Because you are a US citizen (correct?), your petition is approvable immediately...but, plan on it taking at least 3-4 months to come back approved (and even longer is not unusual).
Once your I-130 petition is approved, your husband can then file for Adjustment of Status. This is done via form I-485. Along with that will be an Affidavit of Support (I-864) in which you must show the government that you have sufficient funds to support your husband (and you contract with the government that you'll ensure he not end up on public assistance...even if you and he divorce).
Your husband is permitted to Adjust Status (to a Lawful Permanent Resident) in the U.S....without leaving. This is so even though he's overstayed his visa.
Again, this is the very BASIC outline of what you can do to help your husband. Within the whole process, there are pitfalls that must be avoided. Because this process can be quite convoluted and at times complicated, I really recommend contacting an immigration lawyer to assist you. It will cost more now, but isn't the investment worth it?
Assuming you are a U.S. citizen, you can file for adjustment of status in a 1-step because your husband entered lawfully. Contact an immigration attorney directly for more assistance if needed. Good luck.
Gunda J. Brost
Brost Law Office
This advice does not form an attorney-client relationship and is merely informative. It should not by itself be relied upon to address a legal concern.