The survey, published yesterday by the Chartered Institute of Purchasing and Supply (CIPS), came as a relief amid a slew of gloomy economic indicators at a time when the UK is in danger of recording its third recession since 2008.
CIPS's services activity index edged up from 51.5 in January to 51.8 last month on a seasonally adjusted basis, moving slightly further above the level of 50, which is calculated to separate expansion from contraction, and signalling a marginal pick-up in the sector's growth rate.
But Vicky Redwood, chief UK economist at consultancy Capital Economics, said: "The services survey is still at a fairly weak level and, on the basis of past experience, is only just consistent with positive growth in services sector output."
Surveys from CIPS in recent days have shown renewed contraction in the UK manufacturing sector in February, and a faster decline in construction.
Ms Redwood said: "A weighted average of the three CIPS surveys suggests it is touch and go whether the overall economy will manage to grow at all this quarter."
CIPS noted its survey had now recorded "modest" growth in UK services activity for two straight months, after a fall in December.
Chris Williamson, chief economist at CIPS survey compiler and financial information company Markit, calculated that these purchasing managers' reports were so far pointing to growth of about 0.1% in the first quarter.
He added: "Given the marginal growth, the direction of the surveys in March will - prove decisive in determining whether the country avoided a slide back into recession."
UK gross domestic product fell 0.3% in the fourth quarter of 2012. Any further contraction this quarter would put the UK in triple-dip recession.
CIPS's services survey showed an acceleration of growth of new business in February, and signalled the rate of employment growth in the sector, while remaining modest, also picked up.
Services companies' overall expectation about their activity levels in 12 months' time rose slightly to a nine-month high. But CIPS cited "ongoing concerns" among firms about public spending cuts and said these worries "served as a restraint on confidence".
Howard Archer, chief UK economist at consultancy IHS Global Insight, described CIPS's services survey as "encouraging" but added: "Service companies face a tough task to build on their apparent modestly improved start to 2013, as they still face an uncertain business outlook and tightening government spending - There are still significant pressures on consumers, which are likely to limit the upside for their spending on services for some time to come."
Economists viewed the services survey as reducing the chances of the Bank of England's Monetary Policy Committee voting for further stimulus, through a rise in quantitative easing from £375 billion, when it concludes its latest monthly meeting tomorrow.
But Mr Archer said this "remains a very close call".