But the Hoops headed into a pressure pot when they arrived in football-crazy Warsaw.
Their Champions League second qualifying round clash with Legia is an early test of the transition which is being undertaken as Ronny Deila picks up the reins handed over by Neil Lennon.
The previous incumbent of the Parkhead hot seat had his own problems stepping up to this particular hot plate in 2010, crashing out of Europe in the formative weeks of his managership, first to Braga then to Utrecht.
As the history books show, it all came good in the end for Lennon, with a last-16 place in 2012-13 the highlight of his four seasons in charge.
But while opponents Legia are respectful of what the Hoops have achieved - they have made it out of the group stages three times in the last nine seasons - they are not overawed.
The Poles sense that, with a new manager trying hard to introduce a new style without injured skipper Scott Brown and winger James Forrest, Celtic are vulnerable.
Their sticky start in the qualifying series in Iceland supported this theory.
Although the return leg at Murrayfield was a significant step up in finishing if not form, the Warsaw team and their fervent fans believe a major scalp is there for the taking.
They will attempt to test Celtic's resolve from the off tomorrow night in the Pepsi Arena. And their attacking midfielder Miroslav Radovic isn't holding back as he waits to welcome the Scots.
The Serbian - who snatched the injury-time equaliser to save their blushes in their home tie against St Pat's then scored a double in the 5-0 rout in Dublin to book their place against Celtic - did not mince his words as he warned: "Day after day we are becoming better and better.
"We scored three goals in Krakow at the weekend. I did not score any of them. The time to score is tomorrow."
Radovic insists this is not empty rhetoric. He has studied Celtic carefully in recent days and believes he knows where they can be exposed.
He explained: "We know the pluses and minuses of Celtic. If we want to win, we have to fight on the pitch like Celtic and play with the ball better than them.
"We have a few players who can create something individually, and we plan to show it.
"I hope the Legia fans will be as fantastic as usual. It can be our day."
New Parkhead boss Deila will demand his players ignore all the mind games and stick to the game plan he has laid down, which means expansive, attacking football played out to a quick tempo and driven by slick passing.
The Hoops did not concede in either game against KR Reykjavik but Deila acknowledges Legia are a step above the Icelandic champions.
THE Polish club's support is as demanding as Celtic's. Starved of any European success for two decades, they are desperate to see manager Henning Berg lead them to the Champions League.
The fact the return leg will be played at Murrayfield, as work to convert Celtic Park back to a football arena after hosting the Commonwealth Games Opening Ceremony, could mean the first leg result carries more significance than would normally be the case.
Certainly, while the Legia players are disappointed they won't get the chance to sample the special Parkhead atmosphere on a European night, they can see Celtic losing some of their traditional home advantage, despite the fact their supporters turned up in numbers at Murrayfield last week for the tie against KR.
The target for the Polish champions will be to grab any kind of lead to take to Scotland next week.
They are determined to ask questions of Deila and his players while they are still in the settling-in period of their new relationship.
Vice captain Jakub Rzezniczakz is confident that Legia have a puncher's chance of landing a knock-out blow to the Scottish champions.
Like Radovic, he has trawled through DVDs of Celtic games past and present and reckons the current side does not carry the quality or threat of their predecessors.
He said: "A few years ago Celtic was a very strong team but right now it's a team that we can beat."
To the outsider, it would seem Legia have little in their recent record to justify such open optimism. But Rzezniczakz would invite you to dig a bit deeper for evidence that the club which dominates Polish football is ripe for success.
THE bullish central defender said: "In recent years we've eliminated Gaziantepspor of Turkey and the Russians Spartak Moscow from the Europa League. And, in my opinion, this Celtic team is not better than those teams."
The Pole may be considered to be making a rod for his own back if Celtic live up to their status as the seeded club in this tie and make it through to the play-offs. Especially when last season Legia lost five of their six Europa League group matches, including a 1-0 home loss to Apollon of Cyprus.
But Rzezniczakz appears happy to hype up his team's chances and accept the increase in expectancy which such confidence inevitably creates.
He admitted: "There is big pressure on us from the fans and the media here in Poland to qualify for the group stage of the Champions League. But we can handle it."
The moment Rzezniczakz and his team-mates must prove they are as good as their word will arrive tomorrow night, when Ronny Deila's Celts will be just as determined to ram those boasts back down their throats.