Andreas Seppi gave Italy a win in the opening match by rallying to defeat Vasek Pospisil 5-7, 4-6, 6-4, 6-3, 6-3.
Raonic then came through with a 6-4, 7-6 (7-4), 7-5 decision that improved the Canadians chances of reaching the semifinals.
The winner of this confrontation will meet either the United States or Serbia in the next round.
Raonic served 25 aces to just three for Fognini and lost just two points on his second serve for the entire match.
Seppi and Pospisil needed 3 hours, 15 minutes to complete the opening match.
Pospisil will team with Daniel Nestor in Saturday's crucial doubles match against Italy's Daniele Bracciali and Paolo Lorenzi.