Federico Chiesa and Matteo Pessina came off the bench to both score in extra-time, while 67th-minute substitute Manuel Locatelli helped turn the game too.
Mancini had weighed up starting Locatelli ahead of Marco Verratti, whom he replaced, and delighted in his substitutes' impact and their potential roles moving forward, with a quarter-final date confirmed.
"They were brilliant and that can be a huge advantage for us," Mancini said.
"The fact we have players that can step in and change the game because they are fresh, it was an excellent performance.
"The players wanted to win at all costs and with the subs they did a good job and we were able to win."
The win set a new national record as Italy extended its unbeaten run to 31 games, surpassing the 30-game streak posted under Vittorio Pozzo between 1935 and 1939.
However, Sasa Kalajdzic's 114th-minute header ended its 11-game run without conceding a goal, dating back to October 2020, ending a run of 19 hours and 28 minutes without conceding.
Mancini insisted Italy did not under-estimate Austria, claiming the match would be tougher than its quarter-final against either Belgium or Portugal next Saturday (AEST).
"We knew there would be potential banana skins in this match and we thought it would be tougher than the quarter-final, they are not as good as teams in quarter-final, but they really make life tough for you, they cause problems," he said.
"We knew we scored in the first half it would have been a different game. We said it would be a match we would have to struggle through to get the win. We had to dig deep."
Italy goalkeeper Gianluigi Donnarumma said any quarter-final opponent holds no fears for the Azzurri.
"It doesn’t matter who we face, we have to keep playing our football and keep going," he said.
"The emotions are extraordinary, we can’t wait until we can play in a packed stadium in front of these fans."