In sweltering conditions in the Armenian capital Yerevan, Edouard opened the scoring for the Hoops with a neat finish from Forrest's wayward delivery on the stroke of half time.
The provider then turned goalscorer with 10 minutes to go, pouncing on a loose ball in the penalty area before placing the ball into the net.
McGregor added a third for the Scottish Premiership champions in injury time when he slid the ball into the far corner to all but secure Celtic's spot in the second qualifying round.
The second leg takes place at Celtic Park on Wednesday, July 18.
CELTIC (3-1-4-2): Gordon; Hendry, Ajer, Simunovic; Brown; Forrest, McGregor, Ntcham (Kouassi 67'), Tierney; Dembele (Sinclair 67'), Edouard (Morgan 77')
ALASHKERT FC (4-1-4-1): Cancarevic; Yedigaryan, Voskanyan, Praznovsky, Daghbashyan; Grigoryan; Dashyan, Yedigaryan (Manasyan 82'), Sekulic, Reis (Simonyan 77'); Nenadovic (Romero 55')