The closely-watched IHS Markit/Chartered Institute of Procurement & Supply (Cips) Purchasing Managers' Index (PMI) recorded a reading of 62.9 for May, up from 61 in April, representing the fastest growth in output for 24 years.
A reading above 50 signals growth. It was ahead of the predictions by analysts who had forecast a 61.8 reading for the month.
Tim Moore, economics director at IHS Markit, said: "UK service providers reported the strongest rise in activity for nearly a quarter century during May as the rollback of pandemic restrictions unleashed pent-up business and consumer spending.
"The latest survey results set the scene for an eye-popping rate of UK GDP growth in the second quarter of 2021, led by the reopening of customer-facing parts of the economy after winter lockdowns."
The report revealed that services firms recorded a sharp and accelerated rise in new order volumes in May, with the speed of recovery the fastest since October 2013.
Surveyed companies said this was buoyed by a turnaround in domestic demand following the reopening of large parts of the economy in April and May.
This largely offset a slight fall in export sales compared with April, which operators said was linked to international travel restrictions and post-Brexit constraints on trade with the EU.
The data also showed the strongest rate in employment growth in six years as the easing of restrictions sparked hiring plans. However, surveyed firms said they had seen pressure on business capacity and staff shortages due to the rapid uptick in demand.
Duncan Brock, group director at Cips, said: "This shortfall in talent meant the best candidates were increasingly in demand and demanding higher wages, adding to the highest inflationary rise in business costs since July 2008.
"We will see more pressure for salary rises as basic living becomes more expensive for everyone, as the hike in prices charged by service companies was the highest since 1996."