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Alfredo Morelos missed a hat-trick of big chances against Alashkert – who played more than 45 minutes with 10 men after James was dismissed – though the Colombian's goal in last week's first leg ultimately proved the difference.
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The Gers were without manager Steven Gerrard, skipper James Tavernier and five others due to a coronavirus outbreak in their camp, but they were on top in the first half at Nairi Marzadasht.
Joe Aribo's curling drive was touched on to the post by Ognjen Cancarevic, either side of Morelos firing over when played through and being denied by the Alashkert goalkeeper from another one-on-one.
Despite losing attacker James to two yellow cards in the space of two minutes shortly before half-time, the hosts started to grow into the game and nearly levelled when Aleksandar Glisic headed over the bar from close range.
The best of the remaining opportunities fell Rangers' way, though, with Morelos volleying off target from 10 yards with his latest chance and Aribo being denied by the woodwork for a second time late on.
Gerrard's side will discover its group-stage opponent in Saturday's (AEST) draw, while Alashkert will now compete in the inaugural Europa Conference League competition.