Fit-again Charlie Adam and Nacho Novo had to make do with places on the bench for an unchanged Rangers, while Lee McCulloch missed out despite being free from suspension because of a calf strain.
Euan McLean replaced the injured Grzegorz Szamotulski in United's goal, while on-loan Leicester striker Eric Odhiambo made his first start at the expense of David Robertson.
It was a cagey start as both sides attempted to find their feet on the sodden Ibrox surface.
But Rangers burst into life with a superb passing movement that led to the opening goal.
However, United ought to have had Naismith more closely marked as the forward headed home from eight yards following Burke's cross.
The match became scrappy, with United having the better of the possession until Burke fired in a 20-yard drive that McLean turned wide for a corner from which Alan Hutton sent a header off the top of the crossbar.
Naismith gave Rangers an early lead
Rangers' second eventually arrived after 39 minutes when Ferguson flicked on a Burke cross into the penalty box and the ball flew into the far corner.
There appeared to be a suspicion of offside as Naismith and Daniel Cousin were between Ferguson and goalkeeper McLean.
But, after consulting his assistant, referee Mike McCurry allowed the goal to stand.
Rangers started brightly after the break and a flowing move gave Cousin a chance to extend the lead.
But the big striker took the ball too far wide after trying to round the goalkeeper and the chance was lost.
Substitute Boyd was denied when his drive struck his own captain, Ferguson, who then had an effort blocked by McLean.
Boyd also had a penalty saved by McLean after a push on fellow replacement Novo by Willo Flood.
McLean produced two more fine stops to deny Boyd and Ferguson as Rangers finished in impressive style in the Ibrox mud.
But two goals were enough to give manager Walter Smith a happy start to the new year.
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