The move, which will see hourly pay rates go up to £7.15 from the current minimum of £6.36, is expected to cost the council about £310,000 a year.
The plans were voted through by Labour and the SNP.
The Tories said higher pay costs could see the council's trading arm lose business, in turn leading to job cuts.
The move was also opposed by Lib Dem and Independent councillors.
A motion not to implement the living wage was proposed by Conservative Peter Duncan.
Speaking after the decision, Mr Duncan said: "This move will cost the council significantly - over £300,000 - when our budgets are stretched like never before.
"It will result in job losses, and to cap it all, hardly any of our employees will be any better off at all.
"Almost all low-paid workers are entitled to tax credits, therefore any additional earnings will result in less tax credits being paid."
Labour Group leader Ronnie Nicholson said the price was worth paying.
"It was a victory for the lowest paid workers in Dumfries and Galloway Council.
"Dumfries and Galloway hold the shameful record as Scotland's lowest paid region, with almost 25% of local employees earning below a living wage."
The first part of the process will be the introduction of a living wage supplement from 1 October.
In the longer term, officials are to undertake a wider review with the unions in order to make the living wage part of the council's pay and gradings structure.
Mr Nicholson said the council wanted to lead by example and encourage other employees to adopt it.
He added that plans would be considered to ensure outside contractors used by the council also paid their employees the living wage.
A living wage is seen as the lowest pay rate required to help someone meet their basic needs.
The legal minimum basic wage for workers aged 21 and over is £5.93, increasing to £6.08 on 1 October.