The result means Canada can progress to the World Group semifinals after Sunday's reverse singles, when Milos Raonic takes on Andreas Seppi and either Pospisil or Jesse Levine will face Fognini.
The Canada-Italy winner will play the victor in the U.S.-Serbia series. Serbia leads that 2-1.
In the deciding set, Canada had break opportunities in the third, fifth, 13th and 19th games and finally broke in the 27th with the Italians serving at 13-13. Then Pospisil served out the match.
Italy team captain Corrado Barazzutti had initially nominated Bracciali and Paolo Lorenzi to play the doubles, but after the teams split Friday's opening-day singles he decided to substitute Fognini for Lorenzi.
Canada has never been in the World Group semis since the current Davis Cup format was inaugurated in 1981. However, Canada did reach the semifinals in 1913, losing to the United States.