Liam Kelly had a goal disallowed for offside for Killie before Andrew
Little blasted Rangers' opener on 25 minutes.
Steven Naismith's composed finish put the visitors further ahead just
after the hour mark.
The hosts created several chances and almost reduced the deficit when
Kelly's volley struck the post.
Rangers join Aberdeen, Celtic and Motherwell in Friday's semi-final
draw.
Gregg Wylde earned his first Rangers start in a much changed line-up
for the visitors, with fellow youngster Little also included. Kyle
Lafferty was due to start but pulled out just before kick-off because of
illness and Naismith stepped in.
Kilmarnock made only one change with Mehdi Taouil replacing David
Silva and the Moroccan caused Rangers problems in the early stages.
And the home side thought they had gone ahead when Kelly headed into
the net from Craig Bryson but the offside flag curtailed the home fans'
celebrations.
After a slow start, Rangers carved out an opportunity from a corner
when Gregg Wylde's delivery was headed towards goal by David Weir and
keeper Cammy Bell stooped to clutch the ball as Naismith waited to
pounce on any rebound.
Naismith had been isolated for much of the opening period but when
Rangers did get some support to their main striker, they took the lead.
A long ball upfield was nodded on by Naismith to Little, who arrowed
a volley into the top corner from the edge of the box.
Killie tried to get level and Kelly crossed for Manuel Pascali but
Neil Alexander, in for regular keeper Allan McGregor, claimed the header
shortly before the break.
The Ayrshire side started the second brightly and Conor Sammon's
deflected shot spun wide.
Pascali then had two chances in quick succession; the first, a
header, tipped over by Alexander before the Italian shot wide from the
just inside the box.
Taouil was off target with two efforts as Rangers continued to come
under pressure.
But, after weathering the storm, Rangers doubled their lead against
the run of play as Wylde centred and Naismith controlled with his right
foot before finishing with his left.
Kilmarnock went back on the attack and Sammon's shot cleared the
crossbar before Kelly's 25-yard volley rebounded off the near post.
Rui Miguel, who had come in for Pascali, crossed into the Rangers
six-yard box and Sammon's shot into the turf bounced over the bar.
Vladimir Weiss fired wide of the Killie goal and, at the other end,
Miguel lifted a shot over from a promising position.
Jamie Hamill was next to have a shot at the Rangers goal and, though
the midfielder's attempt beat Alexander, it slipped past the post in the
closing seconds.
