That was up from 5.5% in August, and 5.2% in July. It brings the average house price to £286,000.
However, the rise is still much lower than a year previously, when prices were rising by more than 12%.
The Halifax has previously said that house prices rose by 8.6% over the same period, while Nationwide said the rise was just 3.8%.
Separately, the Land Registry for England and Wales announced that prices were up by 5.6% in the year to October.
The Land Registry includes cash sales, which typically represent about a third of the market.
The ONS figures show that prices rose fastest in Northern Ireland (10.2%) and the East of England (8.4%).
What is the difference between the various UK house price surveys?
John Hawksworth, the chief economist at PwC, said house price inflation was now running at twice the rate of average earnings.
"The ongoing rise in house prices reinforces our projections that, by 2025, only around a quarter of 20-39 year olds in England may be owner occupiers, compared to around three quarters of over-55 year olds," he said.
Jeremy Leaf, a former chairman of Rics, said that, far from running out of steam, the housing market was re-energising itself.
"With the average property price in London now £531,000, unless you earn way above the national average salary, you have precious little hope of being in a position to buy," he said.
"Generation Rent is being left out in the cold: they have aspirations to buy but are being pushed further away from their goal."
House prices rose by 6.1% in the year to September 2015, according to the Office for National Statistics (ONS).