The victory completed a very good week for the hosts, following Monday's 8-1 thrashing of Clyde.
However, Dumbarton were first to threaten, Colin Nish testing Steve Simonsen with an early volley which the keeper smothered without too much difficulty.
However, Rangers quickly took control and Kris Boyd shot just wide after good work from Darren McGregor.
And the hosts grabbed the lead when Lewis MacLeod sent over a corner and Lee McCulloch (15) was on hand at the near post to volley the ball high into the net.
The hosts were extremely unlucky not to increase their advantage, hitting the post twice within the space of three minutes shortly after opening their account.
Youngster MacLeod hit the woodwork from long-range with a low shot and Nicky Clark then shot against the other post after getting on the end of a Boyd knock-down and turning in the box.
David Templeton was then denied by Danny Rogers but the second goal eventually arrived on the half-hour mark when Rangers worked the ball nicely, McCulloch playing in the overlapping McGregor, who rifled his first goal for the club past Rogers at the near post.
There was no respite for the overworked Rogers, who was soon called into action to deny Boyd again, but Nish then went close at the other end, his angled drive drawing a fine diving save from Simonsen after Jordan Kirkpatrick had created the opportunity.
Simonsen did well again to keep out a Turner header, although the Dumbarton man had been flagged offside, and Rogers also distinguished himself again just before the break as he foiled Nicky Law.
Law caused problems again soon after the break, doing well to make a chance for himself and sending a half-volley towards the top corner which Rogers brilliantly saved one-handed.
But Rangers were looking very dangerous going forward and it was 3-0 when Clark (54) shot home after Boyd had done well to cut the ball back.
And there was further misery for Dumbarton when Dean Shiels sent over a corner, Rogers missed it, and the ball bounced into the net off the face and boot of the unlucky Chris Turner (76).
Boyd then had a headed goal ruled out for offside before there was a second own goal, this time at the other end, when Bilel Mohsni (81) headed a curling Archie Campbell cross past Simonsen to give the Dumbarton fans something to enjoy.
However, the final incident of note once more featured the impressive Rogers, who produced another good save to stop Clark from making it 5-1.