Ireland will claim a second straight title if they defeat England later on Saturday thanks to the Italians winning a breathless match in which Juan Ignacio Brex, debutant Louis Lynagh and Stephen Varney all scored tries.
Further kicked points from Paolo Garbisi and Martin Page-Relo ensured a historic win for Gonzalo Quesada's team who moved off the bottom of the table and up to fourth on seven points.
Italy hadn't won at home since beating Ireland in 2013 but came out on top in a match that will live long in the memory of the country's long-suffering rugby fans, who roared with delight after a nail-biting finish.
Saturday's win could be crucial for the battle to avoid a ninth straight wooden spoon as Wales, who are now in last place, face France in Cardiff on Sunday.
Italy also travel to Wales, where they last won a Six Nations match two years ago, in their final match next weekend.
Gregor Townsend's Scotland meanwhile now have a fight on their hands to finish second in the Six Nations for the first time.
Scotland have not finished higher than third since winning the last Five Nations in 1999 and their hopes of ending this year's edition in the top two have been damaged as both England and France can overtake them this weekend.
