The closely watched Markit/CIPS services purchasing managers' index (PMI) rose to 58.6 in November, up from 56.2 in October.
A reading on the index above 50 indicates expansion in the sector.
The services sector has now been showing growth for nearly two years.
It comes as a similar survey covering the eurozone showed service sector growth in the bloc slowed in November. Markit's eurozone services PMI fell to to 51.1 in the month from 52.3 in October.
The data, taken together with construction and manufacturing output in the currency union, points to economic growth in the eurozone of just 0.1% in the final quarter of the year, Markit added.
Data released by Markit on Monday and Tuesday painted a mixed picture of the UK economy, with manufacturing picking up in November but growth in construction slowing.
Markit said the three surveys combined pointed towards economic growth of 0.6% in the final three months of this year, up slightly from an earlier estimate of 0.5%.
"Faster growth of services activity brings welcome news that fears of a potentially sharp slowdown in the economy look overplayed," said Chris Williamson, chief economist at Markit.
The figures will come as a fillip to Chancellor George Osborne, who gives an update on official growth and borrowing forecasts in his Autumn Statement on Wednesday.
The Bank of England forecasts the UK economy will grow by 3.5% this year, faster than any other advanced economy, before slowing to 2.9% next year.
Markit said the services PMI showed that falling oil prices, which hit a five-year low on Monday, were helping firms to offset the effect of staff costs, as hiring increased at its fastest rate since July.
"There are also signs that wage growth is picking up alongside the improving labour market, which should help boost household incomes and consumer spending," Mr Williamson added.
Last week, official data showed household spending rose at its fastest rate in over four years in the three months to September, while Bank of England data on Monday showed consumers increased their borrowing at the fastest rate since the financial crisis.
Activity in the UK's dominant services sector grew at its fastest rate for a year in November, a leading survey has indicated, helping to alleviate fears that the economy may be slowing.