The former Gloucestershire All Golds coach will fill the role that was left vacant by Neil Kelly, who stepped down at the end of the 2018 season.
Greenwood, a former England international, overcame competition from 42 other applicants to land the post.
The 38-year-old has agreed a one-year deal with the club, with the option to extend.
"It is a one-year deal, with an option to extend," confirmed Rams chairman Mark Sawyer.
"We are expecting him to be there for the long-term, it wasn't a difficult decision to give the job to Lee.
"What I would say is that there were some excellent candidates and the fact that 42 applicants put in for the job, it shows that there is a demand out there for aspiring coaches.
"It was nice to have so many options but we felt at the end of the day that Lee was the best option.
"He has coached a club with limited sources before and been successful and he has the ability to move us forward."
The new Rams chief has had spells, as a player, with London, Huddersfield, Halifax, Leigh and Batley, where he ended his professional career.
After retiring, Greenwood had a short stint with NCL side Siddal.
In 2014 he was appointed as the head coach of the All Golds, where he spent four years before leaving the club after they withdrew from League One.
He has spent the past year coaching at the Halifax Elite Rugby Academy.