A guaranteed living wage of £7.50 an hour has also been pledged for 17,000 of the lowest-paid staff from April 2013.
The living wage deal has been termed as the best outside London by Cosla, the umbrella group for councils.
Although the pay rise falls well below inflation of 2.7%, Cosla said local authorities had gone to the "limits of affordability" given the financial pressures being felt at town halls.
Four-fifths of the 17,000 workers to benefit from the new living wage are women. It is 5p above the rate announced by the Joseph Rowntree Foundation.
More than one in 10 Scots work in local government, making it one of the biggest employers north of the Border.
Unions broadly welcomed the pay offer given that better-paid members, those earning £21,000 a year or more, were told in 2010 that their wages would be frozen until March.
Alex McLuckie, senior organiser for the public sector at GMB Scotland, said he was happy with certain aspects of the offer but wanted more, particularly for workers under the £21,000 threshold.
"We're pleased we've seen an end to the wage freeze. We're really pleased that we're going to see living wages introduced to every council in Scotland," he said.
"Given that we've had a period of three years of no pay increase we still think it's a bit light. After a three-year pay freeze we'd like to see a bit more on the table."
Unions will now consult on the proposals with workers, but the Educational Institute of Scotland (EIS) has criticised Cosla for going public with the offer after meeting trade unions in Edinburgh, but before discussions with teaching unions .
Councillor Billy Hendry, Cosla's human resources spokesman, said: "In very difficult circumstances we are offering our whole workforce a 1% pay increase and, in addition to this, we have bolstered the pay of our lowest-paid workers to £7.50 per hour.
"The offer brings to an end the current pay freeze across Scotland's councils. It shows that councils value their workforce and are doing the right things to support them through tough times.
"Councils have worked hard in very difficult circumstances to put together the best possible deal for their workforce, particularly for those at the lower end of the pay scale."
The current national minimum wage for staff aged 21 and over is £6.19 per hour.
The Office for National Statistics has said the average weekly gross wage for public-sector workers was £564.60, £9 more than those in the private sector.
The public sector had seen a 1.6% rise in weekly wages between April 2011 and April 2012, just above the increase in the private sector.
Unions have claimed the ONS figures "peddled the myth" that those in the public sector were better off, because workers such as hospital cleaners, cooks and porters or home care and residential care workers were employed by private-sector companies.
Unison said the figures were skewed given that the lowest-paid privatised jobs are counted as private-sector workers.
Separate research has found private-sector pay deals are running at an average of 3%, many of them based on two to three-year deals.
Unison Scotland said the Cosla offer would be discussed with members at its conference in Glasgow next week, with employers to be informed in the new year whether the deal would be accepted.
The EIS said the offer had not been put to the agreed pay bargaining forum, the Scottish Negotiating Committee for Teachers. It would not comment on the details.
General secretary Larry Flanagan said: "If Cosla representatives have something to say to teachers on pay, then they should do this through the agreed forum."