Eli Persons shows up as a private in the 73rd Ohio Infantry on the Civil War Soldiers System website at:
http://www.itd.nps.gov/cwss/

This website is the result of a cooperative effort between the Federation of Genealogical Society, the National Park Service, the Genealogical Society of Utah and many other groups. Data on the index part of the site was taken from the index to the Compiled Military Service Record index cards. Therefore, if he appears on the website, he should have a Compiled Military Service Record, which you can order from the National Archives.

You mentioned that he didn't appear in the "official" roster. I'm not sure which roster you're referring to. If you mean the roster of Ohioans in the Civil War 12-volume set, there's any number of reasons -- typo, placed in the wrong unit, name spelled wrong, etc.

But, as I said, since he appears on the CWSS site, he should have a Compiled Military Service Record at the National Archives.

Good luck!