Good play between Nacho Novo and Gregory Vignal saw Novo race clear and cross dangerously along the six-yard line but there were no takers and Dundee cleared, with five minutes gone.
Fernando Ricksen then fired wide of the target from 20 yards with a quarter of an hour on the clock, before the Dark Blues' Garry Brady fired his 25-yard effort just past the post.
A great right-foot cross into the Rangers area by Callum McDonald was just missed by Neil Barrett despite the midfielder's full-length dive in an attempt to get his head to the ball.
Five minutes before the break, Novo raced towards the home goal, riding a couple of tackles on the way, but one bad touch by the striker allowed Dundee keeper Derek Soutar to save at his feet.
Robert Mann's 30-yard free-kick on the stroke of half-time was on target but lacked any real power and was easily saved by Ronald Waterreus.
Rangers cranked up the pressure at the start of the second half and Dado Prso, Vignal and Novo all had half chances, but saw their efforts blocked by a dogged Dundee defence.
Fifteen minutes from time a free-kick from Brady was headed out for a corner, from which Mann headed goalwards - but his effort was booted clear by Ricksen three yards from the line.
Soutar dived to his left to deny Prso who had headed Thomas Buffel's cross goalwards with 12 minutes left.
Eight minutes from time Rangers took the lead when Ricksen's corner from the right was met by Marvin Andrews, who headed past Soutar from five yards.
Two minutes later Rangers made sure of the points when captain Ricksen slotted an easy side foot into the corner of the net after Prso had set him up.