The Azzurri won Euro 2020 -- beating Southgate's side on penalties in the final -- but missed out on last November's World Cup in Qatar after losing to North Macedonia in a playoff.
Euro 2024 qualification Group C pits England against Italy, North Macedonia and Malta, starting with Thursday's trip to Naples, where the Three Lions will attempt to win for the first time since 1961.
England overcame their Euro 2020 disappointment with a resounding 4-0 win over Hungary next time out and Southgate is targeting a similar response as they seek to consign December's World Cup quarterfinal exit to France to history.
"Without a doubt [Italy's recent history is a warning], said Southgate. "I think what we did really well after the Euros was our first game was away in Hungary and we gave one of our best performances, I think.
"Our mentality from the start was really, really good. I don't expect that mentality to be a problem but it is a reminder that these early stages of qualifying are crucial.
"I can remember us scoring a late goal against Poland at Wembley with no fans and people would have taken that victory for granted but it was such an important moment in qualifying. We had several of those in our previous qualification campaigns.
"Although some of those groups looked comfortable in the end, there were plenty of nights where that wasn't the case."
Southgate challenged his players to break new ground while recognising the need to start again at the beginning of another tournament cycle.
"They know the level that's required, they know they have the ability to compete at that level," said Southgate, who could earn his 50th win as England boss in his 82nd match.
"Then the fact that whatever we've done in the past is irrelevant tomorrow night because we have to start again. We have to work hard to qualify again and it is a great fixture for us to get that underway.
"In a nutshell it is the sort of challenge we have got to take on and the type of game we have got to start winning.
"We have [done that] over a period but we've got to consistently try to do that. That said, we haven't won here since 1961 so it is also another bit of history we are trying to break down.
"That is a great challenge for this team because they have knocked down so many of those barriers in the past."