Rangers leapfrogged Aberdeen to move into second spot, with Ibrox manager Philippe Clement insisting that there are more improvements to come from his side.
Vaclav Cerny netted his 10th goal of the season which enabled Rangers to take all there points against Dundee but there were moments of consternation for the Rangers support, particularly in the latter stages of the game.
By that point Rangers had hit the woodwork four times within the game.
They broke the deadlock shortly after the interval when Danilo picked out Cerny, who collected before keeping his composure to round Dens goalkeeper Trevor Carson.
Prior to that Ianis Hagi and James Tavernier had hit the crossbar only Hamza Igamane second-half effort which came off the underside of the crossbar ought to have burst the back of the net after he blasted his effort from the six-yard box.
Cyriel Dessers, introduced deep into the second period, also found hit the crossbar in stoppage time as Rangers looked to kill the game.
Encouraged by the reprieves, Dundee tried to nick a late goal. Tony Doherty’s side enjoyed more possession in the second period but struggled to ask too many questions of Liam Kelly, who had been called into action just minutes before kick-off as Jack Butland left to deal with a personal matter.
Simon Murray caused a few palpitations when his effort had to be cleared off the line in the closing stages by Robin Propper.
Rangers remain eight points off of Celtic, who play Dundee United at Tannadice tomorrow.
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