The Hoops, who recorded a routine victory in the away leg eight days ago, took the lead after just eight minutes at Celtic Park when Moussa Dembele weaved into the opposition box and rifled low into the corner of the net from 14 yards.
Celtic were thoroughly home and dry on 35 minutes, adding a third on the night and sixth unanswered goal on aggregate through James Forrest, who raced clear on the counter and fired low into the far corner of the Alashkert net.
Forrest, Callum McGregor and Ryan Christie all had chances in the second half as Celtic upped the pace despite being a man down, while Alashkert mounted several incisive attacks and enjoyed a late flurry but could not muster a consolation goal.
The Glasgow giants will now face Rosenborg in the second qualifying round of the Champions League but may be left sweating over the fitness of Dembele, who pulled up with an apparent hamstring issue late on.
CELTIC (3-5-2): Gordon; Hendry, Ajer, Simunovic; Tierney, Ntcham, Brown, Forrest (Johnston 76'), McGregor; Edouard (Sinclair 64'), Dembele (Christie 66')
ALASHKERT FC (4-3-3): Cancarevic; Dashyan, Voskanyan, Praznovsky, Daghbashyan; Grigoryan, Sekulic (Simonyan 64'), Yedigaryan; Reis (Romero 86'), Nenadovic (Manasyan 45'), Yedigaryan