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The home side started well and almost had the ball in the net within the first few minutes, but Bilel Mohsni's effort was scrambled away in front of goal.
Rangers were eventually made to pay for their wasted chances in the 23rd minute when some poor clearances allowed the ball to be headed through for Stuart Armstrong, who fired in off the post.
Rangers did, however, pull a goal back with just three minutes to go in the half when full-back Stevie Smith fired a 25-yard free kick through the away side's defensive wall, Radoslaw Cierzniak subsequently making a mess of his attempted save and letting the ball trickle into the net.
Dundee United then sealed the victory late in the second half when Nadir Ciftci seized on a defensive mistake to run the ball into an empty net.