The motion was put forward by council leader Ivor Hyslop at last week's meeting of the full council, a move which had been backed by Mid Galloway's Sandra McDowall.
The 22-18 vote margin came despite warnings from the council leader of the impending danger of huge cuts across the country.
He said: "The sooner we take meaningful action, the less painful will be the cuts to services in the long run.
"Taking action now will help to avoid a double hit next year. We can consult with the public over the recess but none of the proposals will be easy.
"We have a responsibility to the citizens of Dumfries and Galloway to take action now and not bury our heads in the sand."
After the meeting, councillor Hyslop added: "It seems that SNP and Labour councillors are in denial at the scale of budget cuts set to hit our local council. Last year officers projected a reduction in our funding of £39 million over three years. It is now clear that the cuts will be even more significant than that, and yet Labour and the SNP seem to have their heads in the sand.
"Our motion would have seen an earlier start made to the additional savings we now require to make.
"The earlier we start, the less painful the changes will be. I'm astonished that opposition councillors seem happy to ignore the inevitable, but in failing to take early action they are making it more difficult to protect the services on which our most vulnerable depend."
The move to cut another £3 million from the council budget was in addition to the £5.5 million from its 2010/11 budget which was agreed in February after a region-wide public consultation.
Labour's Colin Smyth successfully urged members to throw out the motion proposed by Councillor Hyslop and seconded by convener Patsy Gilroy, and said: "We have a dangerous and fragile situation and they want to take £3 million out of the local economy and bring their reserves up to £10m."