The visitors kicked off attacking the River End goal and for the first 12 minutes gave as good as they got without either side creating a real goalscoring opportunity.
After 16 minutes both Paul Furlong and Ray Jones saw their goal-bound efforts blocked.
Norwich then did most of the attacking but although they won free-kicks on either side of the Rangers' penalty area, failed to test Simon Royce with either a shot or a header.
Marcus Bignot was booked for a foul on Darren Huckerby as he broke away down the left wing.
When Lee Croft appeared to be impeded by Mauro Milanese on the other side of the field, referee Phil Dowd annoyed the home fans by awarding a free-kick to Rangers.
After 38 minutes Dublin headed the ball to Robert Earnshaw and his shot brought an acrobatic save from Royce.
Nick Ward replaced the impressive Martin Rowlands and Dublin had a header blocked before Lee Cook was booked for tripping Jurgen Colin.
In the first minute of the second half Jason Shackell headed over from Youssef Safri's corner and Colin fired into the sidenetting.
Adam Drury also got forward and tried his luck but his shot was blocked.
The home side had penalty claims waved away by Mr Dowd when Earnshaw appeared to be pushed off the ball by Damion Stewart.
Most of the play was in the Rangers half but although Norwich won several corners it was not until the 69th minute that Dublin finally broke the deadlock.
He was picked out by Earnshaw with a clever lobbed pass from the left and beat Royce with a low half-volley from 12 yards.
Rangers came back with a flourish, but Croft could have extended the lead when his first touch let him down and the ball ran through to Royce.
Croft then made way for Andy Hughes and Dexter Blackstock came on for Kevin Gallen for the visitors.
Rangers continued to press forward while Norwich failed to catch them on the break, although Earnshaw had a shot blocked and Huckerby fired over.
Stewart then nicked the ball off Earnshaw's toe when he broke away and looked certain to score. In the end it did not matter and the one goal proved enough to net three points for Peter Grant's men.