The sector, which makes up about 80% of the economy, saw many companies working at capacity with full order books.
The services index, compiled by IHS Markit/CIPS Purchasing Managers’ Index (PMI), grew to 54.3 in August from 53.5 in July.
But firms said Brexit worries were slowing investment for the year ahead.
The figures were better than any of those forecast by economists in a poll by Reuters earlier this month.
IHS Markit’s chief business economist, Chris Williamson, said: “Faster service sector order book and employment growth… highlights the extent to which the economy has become more reliant on services to support growth, and in particular an especially strong financial service sector.”
The numbers also make up for the the slowing growth in manufacturing which showed up in PMI figures for August released earlier this weak and was caused by a sharp downturn in exports.
The manufacturing index was at 52.8, the lowest reading in 25 months.
Any reading below 50 means economic activity is shrinking.
The PMI forecast that the economy as a whole will grow at 0.4% in the third quarter of the year, the same as in the second quarter. During the second quarter of the year it received a boost from the World Cup celebrations and the warm weather, according to the Office of National Statistics.
However there had been fears that the rise in interest rates, uncertainty over Brexit and the threat of global trade wars might dampen growth.
Commenting on the services figures, Mr Williamson, said: “[This is] a relatively robust and resilient rate of expansion that will no doubt draw some sighs of relief at the Bank of England after the rate hike earlier in the month,”
“Given the increasingly unbalanced nature of growth and the darkening business mood, risks to the immediate outlook seem tilted to the downside.”
The survey found that businesses were still taking on staff but that confidence for the year ahead slipped to its lowest since March, with businesses saying Brexit uncertainty had made clients less willing to invest, for now.
There are signs that the shortage of labour is driving up some wages. Firms in the PMI survey reported paying higher salaries to recruit hard-to-find staff and reduce employee turnover that was limiting their ability to complete some projects.
Government figures from the ONS show wages, excluding bonuses, grew by 2.7% in the three months to June, compared with a year ago while the official inflation level, as measured by the Consumer Price Index, stands at 2.5%.