The Office for National Statistics (ONS) said the fillip had more than reversed the 0.4% fall in August.
It also reported that sales were 0.6% higher than September last year.
The ONS also said sales made over the internet continued to climb. Non-store retailing, including online purchases, rose by 15.5% over the year as a whole.
“September’s retail sales figures are a bit stronger than we and the consensus had expected, but hardly suggest that conditions are back to normal on the high street,” said Samuel Tombs, UK economist at Capital Economics.
The news comes after inflation figures, released earlier this week, saw it reach a four-year high.