The 44-year-old, the most-capped Romanian player with 134 internationals, replaces Russian Fyodor Shcherbachenko, who was sacked last month following a string of poor results.
Premier League debutants Mordovia went into a three-month winter break rooted to the bottom of the standings with just nine points from 19 matches.
Munteanu becomes the second Romanian coach in Russia's top flight, joining ex-Romania and Chelsea defender Dan Petrescu, who has revived Dynamo Moscow since taking charge of the club after it lost five games at the start of the season.
Munteanu has coached only Romanian clubs until now, guiding unfancied Otelul Galati to their first league title in 2011 before joining Dinamo Bucharest last month.
However, he quit after only four games, citing his desire to work abroad.