Having missed out on a place in the group stages of last season's Champions League, there is an extra determination among those connected with Celtic to ensure that the drought does not extend into a second campaign.
Once famed for producing a string of adventurous goal celebrations, there was very little for the current crop of Stjarnan players to smile about at Parkhead last week.
In fact, the Icelandic outfit's task in the second leg could have been even more uphill had it not been for Celtic striker Leigh Griffiths, who failed to convert his second-half penalty.
Worryingly, they are unable to boast particularly positive domestic form heading into this contest. Having won the Urvalsdeild title for the first time in their history last term, Stjarnan's defence of that trophy has been far from impressive so far.
They find themselves in sixth position with 12 matches played, having won, drawn and lost on four occasions. Their most recent outing came at home to IA on Saturday afternoon - a game that saw Olafur Karl Finsen open the scoring for the hosts during the first half, only to be pegged back by Gardhar Bergmann Gunnlaugsson's 76th-minute equaliser.
However, the Blues can at least point to a positive European home record, having beaten both Bangor and Motherwell during last season's Europa League qualifiers. Inter Milan also travelled to the Samsungvollur 12 months ago and, despite heading back to Italy with a 3-0 victory, they were made to work hard by their hosts.
Were Stjarnan to overturn the two-goal deficit and further defy the odds by reaching the group stages, they would become the first Icelandic side to have ever achieved that feat.
Recent form (all competitions): LLWLLD
His side may be big favourites to advance, but Celtic boss Ronny Deila conceded that he was slightly disappointed by his team's failure to put the tie to bed in the first leg.
"Two-nil is a good result but I hoped we could finish them off. We had chances to win more but we were not effective enough," he told the club's official official website.
"In the end, I'm a little bit disappointed that we didn't score more. But it's good we have a clean sheet and two goals to go on. We will have a good preparation for next Wednesday and we have a very good chance to make the next stage."
Deila was also critical of strikers Stefan Johansen and Griffiths, who argued over which of them would take a penalty, which the latter eventually saw saved by Gunnar Nielsen. Griffiths has since bounced back from that disappointment thanks to the hat-trick that he scored during a 4-1 friendly win over Eibar on Saturday.
While their advantage in this tie may be a healthy one, Celtic need to be wary of their recent Champions League record on the road. Dating back to September 2003, The Bhoys have lost 27 of their 37 away matches, winning on just five occasions.
At the same stage last year Celtic were comprehensively beaten by Legia Warsaw over two legs, only to be reinstated soon after the Poles were found to have fielded an ineligible player.
Recent form (all competitions): WWLWWW
A broken toe ruled Kris Commons out of the first meeting and it could well prevent the 31-year-old, who missed the recent friendly against Eibar, from featuring in this encounter.
In fact, Deila could well be tempted to stick with the same players that started in Glasgow a week ago.
However, there may be a start for Saidy Janko following his recent arrival from Manchester United, while Griffiths is also expected to be pushing for inclusion after his hat-trick at the weekend.
As for the home team, former Motherwell goalkeeper Nielsen is expected to retain his place between the posts, with Jeppe Hansen once again featuring as a lone centre-forward.
Stjarnan possible starting lineup:
Nielsen; Bjornsson, Gudjonsson, D Laxdal, Arnason; Bjornsson, Praest, Johannsson, Bjorgvinsson; Finsen; Hansen
Celtic possible starting lineup:
Gordon; Janko, Boyata, Mulgrew, Izaguirre; Brown, Bitton; Mackay-Steven, Johansen, Armstrong; Griffiths
Head To Head
These two sides had never met each other before last week, but it was not the first time that Celtic have ever taken on Icelandic opposition.
On a positive not for the Scots, their two previous visits to the Scandinavian country have ended in victory. Back in 1975, Kenny Dalglish was among the goals in a 2-0 win over Valur Reykjavik, while in 2014 KR Reykjavik were defeated 1-0.
Meanwhile, Stjarnan played host to Scottish opponents for the first time when Motherwell visited last year, which ended in a 3-2 win for the home team.
We say: Stjarnan 1-2 Celtic
While Celtic's away record in the Champions League does not make for pleasant reading, they should have more than enough to get a result that will send them through. They wasted a number of chances in the first leg and with Stjarnan required to push forward in search of goals, there should be room for Deila's men to exploit.