The Europa League third qualifying round first leg tie, played at the 20,000 capacity Municipal stadium in Drobeta-Turnu Severin, was not the most glamorous of comebacks for Milan and was a far cry from their heyday.
But the win still provided a morale-boosting start to the season for the seven-times European champions.
Switzerland full back Rodriguez, one of four new signings to make his debut alongside Fabio Borini, Franck Kessie and Mateo Mussachio, curled in a trademark free kick one minute before halftime.
It was not all plain sailing for Milan, who have splashed out around 190 million euros ($221.86 million) on new signings so far in this transfer window in their attempt to bounce back from several seasons of midtable mediocrity.
Craiova, making their first European appearance since being refounded in 2013, had the first real chance when Ionut Mitrita forced a full-stretch fingertip save from Milan goalkeeper Gianluigi Donnarumma.
At the other end, 19-year-old Patrick Cutrone, making only his second Milan appearance, saw his shot strike the inside of the post, roll along the line and hit the other upright.
Craiova had several chances to equalise in the second half and finished the match on the attack.
"We couldn't expect too much this evening," said Milan coach Vincenzo Montella.
"Craiova are ahead of us in fitness terms and they have some quick players. It was predictable that we would suffer and it is something we are going to have to do during the season."
"We are still a team under construction. We had a lot of new players, or players who have barely played for us."
In other matches, second-half goals from Domenico Criscito and Aleksandr Kokorin gave Zenit St Petersburg a 2-0 away win at Israeli side Bnei Yehuda Tel Aviv.
Croatia's Osijek managed a shock 1-0 win away to Dutch side PSV Eindhoven while former European champions Red Star Belgrade beat Sparta Prague 2-0 at home.
($1 = 0.8564 euros)
(Writing by Brian Homewood; Editing by Toby Davis)
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