Having drawn 1-1 with Sweden on matchday one, Ireland were dispatched 3-0 by Belgium in Bordeaux to leave them bottom of Group E on goal difference, meaning they require a win and the Swedes to drop points if they are to have a realistic shot at the last 16.
The omens do not favour O'Neill: Ireland are now seven games without a win in European Championship finals, a run stretching back to 1988, and they have never managed to score more than once in a single match at the tournament.
But the 64-year-old insists his players must believe in their chances of causing an upset against Italy in Lille and of giving themselves a chance of qualifying as one of the best third-placed teams.
"I always said from the outset that we would have to win a game in the competition to give ourselves a proper chance of going through to the next round of the tournament, and that is still the case," he told the Football Association of Ireland's official website.
"Obviously there is disappointment immediately after the game because we were upbeat going into the match following the performance against Sweden.
"But now we have a really great game against Italy to look forward to, to put our immediate disappointment to the side, and if we can win this game we give ourselves an incredible chance of qualification."
Italy have responded to pre-tournament doubters in resolute fashion so far, with coach Antonio Conte having masterminded a 2-0 win over Belgium in their opening match before Eder's late goal ensured a 1-0 victory against Sweden that guaranteed top spot in Group E.
Despite resounding success so far, Conte was unhappy to see a lack of blue shirts in the stands in their last outing and has urged Azzurri fans to show their colours and voracious support in the meeting with Ireland.
Wing-back Matteo Darmian has echoed his coach's rallying cry, telling Euro 2016's official website: "We know that the Ireland fans are a very lively bunch. They make their voices heard a lot.
"But our fans will make themselves heard too. As the coach has already done, we also want to appeal to the fans to unite behind the Azzurri shirt."
Italy, unbeaten in 12 competitive games under Conte, are likely to take the opportunity to rest some of their senior stars.
Antonio Candreva is a doubt with a hamstring problem, while captain Gianluigi Buffon has been nursing a cold and could be left out.
For Ireland, Jonathan Walters (Achilles) is battling to be fit enough to feature in attack as his side look to inflict upon Italy only a third group-stage defeat in 24 games at the European Championship.
Key Opta Stats:
- This will be the 14th meeting between these sides; Italy have won eight of the previous 13 while Republic of Ireland have won two (D3).
- The only previous time Italy won their first two Euros games was in 2000, when they eventually reached the final.
- Italy have never scored more than two goals in their 35 games at the European Championship.
- They have kept a clean sheet in each of their last four games, their longest run since the 2006 World Cup, which they eventually won.
- In their last game, Republic of Ireland failed to have a shot on target in a major tournament game for the first time since the 1994 World Cup (versus Norway).