Goals from Emanuele Giaccherini and Graziano Pelle helped the Azzurri to maximum points in their first Group E fixture at Stade de Lyon on Monday.
Little was expected at the European Championship from an Italy side lacking the quality of their predecessors.
And, although the result against Marc Wilmots' star-studded Belgium has altered some perceptions of his team's capabilities, Conte knows it will take something special for them to go all the way in France.
"I think I've got a very intelligent group of players at my disposal," he said. "All of whom are aware we've achieved something significant, because we've put ourselves in a good position to go through the group stage.
"They know we have to work and give 100 per cent. Because if we're simply content to produce the ordinary, that's not good enough. We have to do something extraordinary to make people proud of us."
Visibly irked by repeated references to his team being deemed one of the worst Italy sides in living memory prior to the competition, the former Juventus boss made it clear after the victory that his players and the coaching staff do not take their privileged position for granted.
"We feel a great sense of responsibility," Conte said.
"I was very emotional... being head coach of the national side at a major championship is not an everyday occurrence. I think the players showed how much the national team means to them despite the difficulties we've come across."
Next up for Italy is a match in Toulouse on Thursday against Sweden, who drew their opener with Republic of Ireland>