The Romanian ATR 72 was carrying 50 passengers when it arrived in Rome Saturday, Italy's ANSA news agency reported.
A passenger on the flight told ANSA the plane attempted to touch down twice before the landing gear bent and it veered off the runway.
One passenger was hospitalized with a spinal injury that was non-life-threatening while two others suffered pelvic injuries, ANSA said.
After the incident Alitalia said "strong winds" forced the plane off the runway. An investigation into the incident was under way.
The consumer group Codacons Monday called on Alitalia to explain what happened as several incidents have raised questions about the Italian airline's partnership with the Romanian carrier.
Last month a Carpatair flight from Ancona to Rome had to make an emergency landing shortly after takeoff and in December a Carpatair flight out of Pisa was stopped just before takeoff when ground crews noticed it was leaking fuel.
Carpatair, the largest airline in Eastern Europe, and Alitalia have a code-sharing agreement and the Romanian airline uses Alitalia livery on its ATR 72s.