Vicky Redwood, chief UK economist at consultancy Capital Economics, cited "tentative hope" that a triple-dip recession might be avoided in the wake of yesterday's service sector survey from the Chartered Institute of Purchasing and Supply (CIPS).
However, she emphasised there were still few signs of an actual recovery.
CIPS's business activity index for the UK service sector rose from 48.9 in December to 51.5 in January on a seasonally adjusted basis, moving back above the level of 50 calculated by CIPS to separate expansion from contraction.
The December fall in services activity was the first for two years on CIPS's measure.
CIPS's services survey, compiled by financial information company Markit, does not include retail. Industry figures this week showed year-on-year growth in UK retail sales value accelerated sharply to 3% in January, from 1.5% in December. This year-on-year sales growth is the fastest reported by the British Retail Consortium since last September.
However, Ms Redwood said: "Before we get too carried away, all the improvement does is suggest the economy is not contracting any more."
CIPS last week published a survey showing UK manufacturing output rose last month at its fastest pace since September 2011. However, manufacturers suffered a sharp slow-down in new order growth and employment was virtually static. A separate survey from CIPS on Monday showed UK construction output fell for a third consecutive month in January.
Ms Redwood calculated that a weighted average of CIPS's surveys was consistent with UK gross domestic product (GDP) "broadly stagnating".
She said: "We continue to expect the economy to struggle to grow by much this year, reflecting the combination of the Government's austerity measures, another year of falling real pay for households and economic weakness in the UK's biggest export market (the eurozone)."
Data published last month by the Office for National Statistics showed UK GDP fell 0.3% in the fourth quarter of 2012. The UK's third recession since 2008 would occur if GDP were to fall again in the first quarter of 2013.
CIPS's latest services survey showed growth in new business resumed in January. It pointed to a modest rise in service sector employment in January, after a slight dip in December.
However, it also highlighted pressure on service sector firms' profit margins, with the prices they charge rising at a much slower pace than their costs.
Although some respondents voiced worries that public-sector spending cuts would hit their performance, companies were at their most confident for eight months about prospects for a rise in business activity over the coming year.
Howard Archer, chief UK economist at consultancy IHS Global Insight, said: "With the improved services sector survey coming on top of the BRC reporting surprisingly decent retail sales in January, it appears that the negative impact of the snow on the economy was limited and it is seeing modest underlying growth at the start of the first quarter. The risk of a triple-dip recession appears to have receded, temporarily at least."
However, he added: "It remains to be seen whether service companies can build on their improved start to 2013, as they currently still face an uncertain business outlook and tightening Government spending. In addition, there are still significant pressures on consumers."