Total UK unemployment fell by 64,000 to 1.49 million.
According to the Office for National Statistics, average pay was up just 1.8 per cent year-on-year in the month.
A bright spot was that weekly average earnings excluding bonuses rose to 2.0% from 1.8% a month before, better than the consensus estimate of 1.9%.
Inflation marched to its highest level in almost four years at 2.9% in May, with the Bank of England expecting the cost of living to peak at 3% by the autumn.
Other rate-setters, though, are likely to point to the continued weakness of wage growth as an indication that the United Kingdom economy remains far from operating at capacity.
When the impact of inflation is factored in, real weekly wages fell by 0.5 per cent compared with a year earlier.
Those in work climbed to around 32 million, a rise of 324,000 on previous year and the largest total since records began in 1971.
Britain's headline employment rate grew to 74.9% in the month, a record high since comparable data was first compiled in 1971.
It meant the unemployment rate fell to its lowest since 1975, at 4.5%, the Office for National Statistics (ONS) added.
That figure is forecast to turn increasingly negative as consumer-price inflation pushes past 3 per cent, driven by the fall in the pound since the Brexit vote.
Bank of England policymakers have said they are watching nominal pay growth data carefully for signs of inflationary pressure.
"The data certainly fuel suspicions that the Bank of England will be in no hurry to raise interest rates; there's a strong argument that hiking borrowing costs at a time of falling real pay and heightened uncertainty regarding the economic and political outlooks would merely introduce another headwind to an already-struggling economy", Williamson said.
Focusing on employment, the ONS said those classed as economically inactive fell by 57,000 in the three months to May to 8.83 million.
The squeeze on living standards is sapping consumer spending, the engine of the British economy, yet a three-person minority pushed for an increase last month to curb inflation.