Goals from Fernando Ricksen and Stevie Thompson saw Rangers through to the CIS Insurance Cup quarter finals.
In first half stoppage time Ricksen curled a wonderful 20-yard free kick beyond David Preece.
Aberdeen pushed forward in the second period and Scott Severin, Scott Muirhead and Derek Adams went close.
But, with two minutes left, Thompson found room in the penalty area to prod the ball home before Bob Malcolm was ordered off for a second booking.
Rangers started sloppily and Stefan Klos had to bail Craig Moore out with a block to deny Darren Mackie in only the second minute after the striker had intercepted a slack pass.
The visitors first threatened when Thompson did well to turn Dons' skipper Russell Anderson and fire just wide from 25 yards.
Aberdeen opened up Rangers in the 18th minute when Steve Tosh's cross found the unmarked Richard Foster eight yards from goal.
The 19-year-old made a hash of his header but, having sent it away from goal, saw it bounce off Boumsong and into the path of Steven Craig, who could only fire wide.
Within seconds of the interval Rangers were ahead with Ricksen netting direct from a free-kick.
With an hour gone Aberdeen midfielder Severin was denied by Klos when he met an Adams corner with a thumping diving header.
There was another escape for Rangers in the 66th minute when Tosh robbed Gregory Vignal and sent a cross into the box that Moore inexplicably allowed to run beyond him.
Substitute Muirhead seized on to the loose ball but could only fire across goal and wide from a tight angle.
Adams then curled a clever effort narrowly over the crossbar.
Ricksen tried his luck from another free-kick after Anderson had tugged Shota Arveladze back but fired straight at Preece.
But the goalkeeper spilled it and had to scramble to regain control before Arveladze could pounce.
Nacho Novo was sent on with three minutes remaining and had a hand in Thompson's clincher 60 seconds later from another free-kick move started by Ricksen.
It was a fine finish from the striker, who marked his first start of the season with a well-placed effort into the corner.
There was still time, however, for Malcolm to be sent off, with his second bookable offence a stoppage time foul on Steve Tosh.
Aberdeen: Preece, Hart, Anderson, McGuire, McNaughton, Tosh, Severin, Adams, Foster, Craig (Muirhead 44), Mackie. Subs Not Used: Esson, Morrison, Tiernan, Considine.
Booked: Foster, Adams, McNaughton.
Rangers: Klos, Khizanishvili, Moore, Boumsong, Vignal, Burke, Malcolm, Ricksen, Vanoli, Thompson, Arveladze (Novo 88). Subs Not Used: Andrews, Lovenkrands, Hughes, Graeme Smith.
Sent Off: Malcolm (90). Booked: Malcolm. Goals: Ricksen 45, Thompson 89.
Referee: K Clark