Their first ever European tie was against St Johnstone in a Europa League qualifier in July 2015.
They knocked the Saints out on the away goal rule in the first round of qualifying and progressed to play Aberdeen's eventual conquerors Kairat Almaty in the next stage.
They recorded a 2-1 home win over the Kazakh side but were eliminated after they fell to a 3-0 away defeat.
In total, the Yerevan-based outfit have played 12 matches in Europe – eight in the Champions League and four in the Europa League – and have won all their first-round ties in both competitions.
As well as St Johnstone and Kairat Almaty, the Armenian side have faced Santa Coloma of Andorra on two separate occasions, Dinamo Tbilisi of Georgia and Belarussian side BATE Borisov. They recorded home wins over St Johnstone (1-0), Kairat (2-1), Santa Coloma (3-0 and 1-0).
The Armenian champions play their home games at the 6850-capacity Alashkert Stadium but the match has been moved to the much larger Republican Stadium.
The hosts are attempting to fill the national ground, which can hold around 15,000, by giving away free tickets.
According to the club's chairman Bagrat Navoyan, their own stadium will be either renovated or entirely rebuilt by 2020, with a possible expansion of the capacity up to 10,000.
Brendan Rodgers will go up against Varuzhan Sukiasyan – a manager with vast experience in his home country.
He has managed over ten clubs in Armenia as well as having two spells in charge of the national team.
Sukiasyan won the first of his Armenian Premier League's in 1998 with Tsement Ararat and this year won the title with Alashkert by six points.
Between 2010 and 2015 Sukiasyan worked as the chief of the Yerevan Funeral Home in the country's capital.
The Armenian champions have recruited five new players ahead of the Champions League first round tie, one of whom has a reputation as a hothead.
Brazilian attacker Jefferson Reis saw red earlier this year whilst playing for Operaio after he battered a ball boy for excessively celebrating a goal against his team. He is expected to start on Tuesday.
And one of Alashkert's players has actually spent time with Celtic – a week to be exact.
Defender Dino Skvorc, spent seven days training with then-manager Neil Lennon at Celtic in 2010 when he was a Croatia Under 21 prospect but the Parkhead side reportedly weren't prepared to pay the £1m fee
Skvorc went on to play in Israel, Romania and Cyprus before signing for Alashkert earlier this year.
Armenian international striker Artak Yedigaryan could prove to be a potential danger after finishing top scorer in the league for the last two years.
Brendan Rodgers' side face an uncomfortable evening in the Armenian capital with the temperatures likely to be around 33°C at kick-off.
Celtic could face a rematch with Rosenborg should they overcome Alashkert.
The Norwegian side face Iceland's Valur in their first round qualifying tie, with the winner meeting either the Scottish or Armenian champions in the next round.
Brendan Rodgers can take confidence from last seasons meeting where they defeated Rosenborg 1-0 on aggregate in the third qualifying round.