Rodriguez's 21st-minute effort followed a 15th-minute opener from Edinson Cavani. He went on to play 82 minutes in his first appearance alongside Cavani, Luis Suarez and other members of Tabarez's core group.
Sporting Kansas City's Botond Barath was also in Hungary's backline for the opening 72 minutes.
Rodriguez previously scored against the US national team in September and against Peru in October. Uruguay are next in action Tuesday in a friendly at rival Argentina.