Eleven years later, Neil Lennon hopes his team can retrace their steps to the Champions League.
The 3-1 victory in the opening qualifier of 2001-02 was a seminal moment in the development of the team O'Neill had built with a view to re-establishing the Parkhead club on the European stage. At the heart of his side was Lennon, who made his European debut for the club against Ajax, who included a young Zlatan Ibrahimovic and Rafael van der Vaart in their ranks, along with Shota Arveladze and Cristian Chivu.
Lennon admits they travelled to Amsterdam more in hope than expectation, but returned with an impressive lead borne of an even more impressive performance. The return leg exposed just how naive this group of players was as Lennon recalls they did not know whether to go for the kill, or play it safe. They lost 1-0 in Glasgow, but made it to the group stage for the first time, and Lennon sees a friendly against the same opponents as a staging post on their way back.
He said: "It was a big step to get us into the Champions League. We didn't know what we were going into. Ajax were a very good side, but we played brilliantly on the night and scored three really good goals.
"We didn't know whether to stick or twist in the second leg. It was a night full of real tension. Thankfully, we got through."
Like the team of 2001, this Celtic team carries an element of surprise, so little exposure have these players had in Europe's elite competition.Lennon does not underestimate how useful that can be.
"We probably caught them a bit cold," he said."They probably didn't know much about us. We definitely surprised them on the counter attack and got two great goals early on, from Bobby [Petta] and Didier [Agathe], then Arveladze scored to put the tie in the balance.We had to withstand a fair bit of pressure but Chris [Sutton] scored a great goal to ease that a bit.
"I can't remember how we were feeling when we left. We were just delighted with the result but knew there was more to come.
"The second leg taught us these games are far from over. It was really, really tense, particularly in the second half, when another goal would have evened the tie up."
The removal of Ajax by unseeded Celtic raised eyebrows across Europe. Within a couple of years, O'Neill's side was contesting the UEFA Cup final in Seville. It is in the Champions League that reputations, and millions, are to be made, however, and Lennon wants his young team to benefit from competing at that level.
"There's a real incentive for them to get through and enjoy the competition," he said. "There's a sense of expectation and excitement going into the groups, something these guys have never had.
"There's no problem with the temperament of the guys here. They will be taking a step up in terms of the quality of opposition, but they're a good bunch, technically. The Ajax game will be a great experience because it will be a good exercise in playing against a team who are very good at keeping the ball."
Should Celtic succeed in eliminating HJK Helsinki, Lennon acknowledges the hard work will only just have begun. However, he is reassured this group of players learned enough in last season's Europa League to be ready.
"It does give you a source of encouragement," he said. "The overall performance in Udinese was very, very decent.
"When we went into the Champions League for the first time, we didn't know what to expect. It was all good and it enhanced you as a player. It made you realise where you were in the game and where you needed to go. That's what I want for these players. I want them to sample it. But I can't stress enough that I'm wary of these qualifiers because of when they are being played."
One of that group, James Forrest, could have been playing for Team GB against Brazil last night, but the 21 year-old midfielder insists he is happier preparing for the new season with his team-mates. An ankle injury sustained during the League Cup final in March killed off the issue of whether or not he would join the British team.
He said: "I always wanted to play in the Champions League so it [playing in the Olympics] wasn't really a hard decision. I was always going to stay with Celtic.
"I was injured for the last two months of the season, and I feel stronger and fitter coming back to pre-season. Hopefully, that helps me over the course of the season.
"If you want to get into the Champions League, you need to be able to play against sides like Ajax, so this will be a tough test for us."
Beram Kayal has been ruled out of tonight's friendly after he damaged his left ankle in Tuesday's game against Vfr Aalen.