The hosts, who finished with nine men, grabbed an early opener through Vahan Bichakhchyan in the sixth minute after a defensive mistake from Grant Hanley but were left to rue a slew of missed chances as Southampton midfielder Armstrong made them pay with two superbly taken goals.
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Scotland were helped by Arman Hovhannisyan's dismissal just before the break and turned on the style in the second half, with captain John McGinn and Che Adams securing the points and easing the pain of their World Cup qualification disappointment.
There was more misery for Armenia as Kamo Hovhannisyan was shown a straight red card in stoppage time for a dangerous challenge on Lewis Ferguson.
Steve Clarke's side are second in League B Group One with six points from three games, behind Ukraine who play Ireland later on Tuesday. Armenia have three points from four games.