In added time at the end of the first half, French striker Odsonne Edouard, who made his loan deal from Paris St Germain permanent in the summer, redirected a wayward James Forrest drive into the net for his first European goal for the Hoops.
Boss Brendan Rodgers was without a clutch of players due to a combination of World Cup commitments and injuries, but Celtic still had too much for their Armenian opponents and Forrest scored a second in the 81st minute with a close-range finish before Callum McGregor added a third in the 90th minute.
The Scottish champions need to negotiate four qualifying rounds this year to reach the lucrative and prestigious group stages for the third year running.
The second leg of the first qualifier is at Celtic Park next Wednesday night where they will, barring a remarkable turnaround, book a place against Valur Reykjavik or Rosenborg.
The champions of Armenia for the past three seasons showed little in the way of an attacking threat until the 20th minute when Artak Dashyan left Celtic defender Jozo Simunovic on the flank and drew a fine save from Craig Gordon at his near post before Uros Nenadovic missed the target seconds later.
Alashkert had switched the game from their own smaller ground to the 15,000-capacity Vazgen Sargsyan Republican Stadium but it was far from full as Celtic remained positive and six minutes later home goalkeeper Ognjen Cancarevic did well to push away a curling shot from Moussa Dembele.
In the 36th minute, Celtic midfielder McGregor dragged a shot inches wide of the far post after a flowing move.
But with three minutes of stoppage time almost complete, Edouard was alert to steer Forrest’s wild volley into the net from 10 yards to swing the tie firmly in Celtic’s favour.
The visitors survived a scare at the start of the second half when Gordon had to make a save from Brazilian Jefferson Reis and then shortly after the hour he tipped a powerful long-range effort from Artak Edigaryan over the crossbar.
The home side powered forward when they could but Celtic remained concentrated for the most part and retained a threat of their own.
Lewis Morgan came on for Edouard in the 77th minute to make his competitive debut for Celtic following his move from St Mirren after Scott Sinclair and Eboue Kouassi had earlier replaced Dembele and Olivier Ntcham.
And with nine minutes remaining the visitors made it 2-0 following a corner, Forrest assuredly side-footing the ball into the net from 12 yards after it had bobbled around the box.
He then set up McGregor for the late third, his fellow Scotland international rifling a shot into the far corner of the net to make the return game all but academic.
Alashkert: Cancarevic, Yedigaryan, Voskanyan, Praznovsky, Daghbashyan, Grigoryan, Dashya, Jefferson Reis (Simonyan 85), Edigaryan (Manasyan 82), Sekulic, Nenadovic (Romero 55).
Subs not used: Avagyan, Zeljkovic, Stojkovic, Antonic.
Celtic: Gordon; Hendry, Ajer, Simunovic; Brown; Forrest, McGregor, Ntcham (Kouassi 67), Tierney; Dembele (Sinclair 67), Edouard (Morgan 77).
Subs not used: Bain, Christie, Allan, Ralston.
Referee: D Grujic