Despite their run, Bonucci said Italy – who claimed their only European Championship in 1968 – still had progress to make.
"We will reach the level of the top teams in Europe but we still need to improve to compete against those teams who have achieved something important in recent years," the Juventus defender told a news conference.
"We are not at the level of Spain, France, Germany and England because of our lack of experience.
"They have a lot of world-class players and we don't, but we have all the qualities and the potential to reach that level."
The great Arrigo Sacchi recently hailed Italy's performances under head coach Roberto Mancini, who took over after the nation failed to qualify for the 2018 World Cup.
Mancini – whose side have won all nine of their Group J qualifiers ahead of facing Armenia – was thankful for the praise and said he wanted Italy to develop a winning mentality.
"Thanks to Arrigo Sacchi for the compliments, it is such a pleasure to receive nice words from someone like him," he said.
"I was not expecting 10 wins, but of course I was expecting qualification.
"Winning games was not our priority at the beginning. Our main task was to do something new and different to make our fans happy and passionate again.
"Then, we wanted to give our players a winning mentality, because this is the best way to achieve success."