With the sides tied at 14-14 heading into the last five minutes, Edoardo Padovani’s penalty looked to have secured a first win in the competition this season for the Italians.
However, Aki touched down following a driving maul at the death to secure an important win in Connacht’s bid to chase down Conference B leaders Munster.
The Westerners trailed their fellow Irish province by nine points heading to Italy, with Munster playing Cardiff Blues later on Friday before a meeting with Connacht at Thomond Park next weekend.