The PMI indicator, which gauges month-on-month changes in combined manufacturing and services output, held firm at its August high of 58.3. Services expanded at the joint-fastest rate in the survey's history, though there was a pullback in the (seasonally-adjusted) rate of growth in manufacturing, from 56.7 to 53.4, the lowest rate since May.
The pace of job creation was also unchanged since August, remaining "solid and well above the long-run average", the survey says. "The degree to which staffing levels rose was also in line with the UK average. Although the service sector led the increase in staffing numbers north of the border, Scottish manufacturers likewise posted a notable rise."
The level of new business at Scottish firms also matched August's series record. September's rise was the tenth in successive months, supported by further growth in new export orders at manufacturers. A strong housing market was another factor driving the upturn, according to reports from survey panel members.
The travel, tourism and leisure sector led the rises in both business activity and employment.
Backlogs of work were accumulated at businesses during the latest survey period, continuing the trend observed in each of the past four months, with the latest increase more marked than in August.
Higher food, fuel and labour costs meanwhile helped lift the rate of input price inflation in Scotland to the fastest in five months, and one that was well above the UK average. But margins remained under pressure as businesses largely absorbed the higher costs, recording on average only a marginal increase in output prices due to a competitive market environment.
Donald MacRae, chief economist at Bank of Scotland, said: "September's PMI showed the private sector of the Scottish economy continuing to expand across both manufacturing and service sectors providing further evidence of the strengthening of the recovery. Output and new business rose at survey-record equalling rates accompanied by growing employment and rising new export orders. The PMIs of the last six months suggest the Scottish economy not only grew in quarter two this year but saw that growth accelerate in quarter three. The recovery is gaining momentum."
Meanwhile the autumn forecast from the EY Item Club published today says the UK economy will grow by 1.4% this year, up from its 1.1% target three months ago and in line with last week's upgrade by the International Monetary Fund. However the report said the UK's short-term growth was being fuelled by consumer spending, not by a rebalancing of the economy towards exports and investment.
The report's author economist Peter Spencer said he expected those to "gradually perk up" in 2014, helping the UK economy to grow by 2.4%, rising to 2.6% in 2015.
But he warned that as disposable incomes would creep up by only 0.2% this year, consumer spending would grow by 1.6%, the gap would have to be filled by savings.
This "needs to be supplemented by a thrust from the engines of export and investment", Mr Spencer said. "Otherwise, there's a real risk the recovery will falter."
Mr Spencer said he believed that despite criticism of the Help to Buy scheme, it would "help level the playing-field for first-time buyers and low equity households", and added: "The chances of seeing another housing market bubble are extremely slim."
The UK government brought forward the second phase of the English scheme last week, while the Scottish version of the scheme launched at the end of last month.