Defeat to Hungary on Thursday had ended hopes of nudging into a play-off place in Group Seven but Noel King's men finished with a flourish despite finishing the game with nine men after two red cards.
Sean Murray put Ireland in front in the 23rd minute but Luca Caldirola equalised on 35 for the Italians, who were already guaranteed qualification as group winners.
Another Italian victory seemed certain when Derrick Williams was sent off in the 37th minute for a professional foul, but Aaron McCarey saved the resulting penalty and instead it was the visitors who took control.
Two goals in as many minutes just before the hour from Doran and Conor Henderson gave the Irish a 3-1 lead and they made it four eight minutes from time when Doran added his second of the game.
Anthony Forde was the second Republic player to see red for two bookable offences in the dying minutes, and Stephan El Shaarawy grabbed a consolation goal for the home side in injury time.