There was also an increase in the number of temporary staff finding jobs, following a modest decline in December.
However availability of both temporary and permanent staff contracted last month, the latest Recruitment and Employment Confederation (REC) Report on Jobs for Scotland found.
The REC warned the struggle to find appropriate candidates will get worse.
It is calling on the government to adopt a “balanced and evidence-based” immigration system, putting the concerns of businesses at the top of their agenda to ensure the success of the Scottish economy.
The study, carried out by IHS Markit, found that growth of demand for permanent and temporary staff in Scotland moderated during January.
Though the number of permanent vacancies increased sharply, the pace of expansion weakened to a one-year low.
Meanwhile, demand for temporary staff rose at the softest pace since August 2016.
Growth in vacancies for both short-term and permanent staff at the UK level was stronger compared with that seen in Scotland.
REC chief executive Kevin Green said: “It’s reassuring that demand for permanent staff is stable despite the economic uncertainties businesses are facing.
“However, we should keep an eye on the slower growth in demand for agency workers. This could be a sign that employers are hesitating.
“The struggle to find appropriate candidates will get worse. We are therefore asking for a balanced and evidence-based immigration system.
“Businesses urgently need to invest in the up-skilling of their workforce. It’s time that the government puts the concerns of businesses at the top of their agenda to ensure the success of the Scottish economy.”
Scottish recruitment consultancies signalled further increases in pay in January.
Starting salary inflation for permanent jobs was steep and quickened to a 37-month high.
However temporary wage inflation in Scotland softened to a 15-month low and was the weakest of all surveyed UK regions.
Mr Green said that now could be a good time to move jobs as the inflation of starting salaries has hit a three-year high.
He said: “We are seeing a continued rise in jobs filled via recruiters in Scotland as it gets more challenging for businesses to find candidates.
“The UK has almost full employment and the country is plagued by labour, skills and talent shortages.
“This increasing competition for good quality staff means that employers are willing to pay higher wages to attract the right people. Starting salaries in Scotland have gone up at the quickest rate in over three years.
“So, it’s a good time to move jobs, especially as employers aren’t increasing wages for their existing workforce.”
The report is based on a monthly survey of around 100 recruitment and employment consultants.
A Home Office spokesperson said: “We are committed on building an immigration system that works in the interests of the whole of the UK, including Scotland.
“Decisions about our future immigration system will be based on evidence, which is why we have asked the independent Migration Advisory Committee to assess the economic and social impact of EU citizens in all parts of the UK.
“We are engaging with and considering the view of all stakeholders - including the Scottish Government and businesses in Scotland.
“Scotland as well as other devolved governments have different ways available to encourage more individuals and families to move to their part of the UK.”