The rate hit 5.1% in the three months to November - its lowest rate since the three months to October 2005, according to the Office for National Statistics.
The number of people out of work fell by 99,000 to 1.68 million in the three-month period.
Average weekly earnings, including bonuses, were up 2%, the slowest increase since February.
The 2% growth in wages was below the 2.1% growth forecast in a Reuters survey.
Excluding bonuses, average weekly earnings growth slowed to 1.9% in the three months, the ONS said.
The figures show that the employment rate hit 74% - the highest since comparable records began in 1971.
The figures come a day after Bank of England governor Mark Carney ruled out an early rise in interest rates because of the turmoil in the global economy and weaker UK growth.
A sustained improvement in wage growth was one of the factors he said would help the Bank to gauge when to raise rates.
He has previously said he would like to see earnings growth at above 3% a year before rates are increased.
Capital Economics economist Ruth Miller said the figures confirmed an interest rate hike "is still some way off".
"There still seems very little inflationary pressure coming from the labour market," she added.
Martin Beck, senior economic advisor to the EY ITEM Club, the forecasting body of accountancy firm EY, said the jobless rate was now below the pre-crisis average of 5.2%, yet it had not triggered excessive pay growth:
"The latest pay numbers will exacerbate concerns raised in recent MPC meetings that very low inflation may be pushing down pay settlements, threatening a negative feedback between wages and prices.
"That the first post-crisis rate hike may be pushed back into 2017 is looking an ever-more realistic possibility."
The latest ONS figures show that almost 23 million people are now in a full-time job, 436,000 more than a year earlier, while 8.4 million are working part-time, up by 152,000.
The number of workers in part-time jobs wanting a full-time post is 1.2 million, down by 21,000 in the latest three months.
Economic inactivity, counting people on long-term sick leave, looking after a relative or who have given up looking for work, fell by 93,000 to just under nine million, the lowest since the spring of 2014.
The inactivity rate for women reached a record low of 27%.
The figures for unemployment in the three months to the end of November are based on the Labour Force Survey, in which the ONS speaks to 40,000 households once a quarter, making it the country's biggest household survey.
The ONS is 95% confident that the figure of a 5.1% rate of unemployment is correct to plus or minus 0.2 percentage points - in other words, it's between 4.9% and 5.3%.

Met Commissioner Sir Bernard Hogan-Howe told an audience at the Royal United Services Institute (RUSI) that firearm officers needed greater public support.
He also revealed that despite a push for more armed officers, there were not enough people willing to do the role.
Sir Bernard is retiring after more than five and a half years in the post.
During the speech, the Met's commissioner said public pressure on firearm officers had meant "we simply don't have enough people now wanting to do these jobs".
"This a dangerous place to be... we can't afford to have officers think twice because they fear the consequences of shooting someone".
"That's how they get shot or the public gets hurt or a criminal gets away with a gun," he said.
Sir Bernard also stated that armed officers did not fire a shot at a suspect in more than 3,300 incidents involving firearms in 2016.
"It's an urban myth that officers are trigger-happy," he said.
The Met has said it will increase firearm officer numbers by 600, with more than 400 extra staff trained or selected to date.
According to a staff survey, nearly half of Met Police officers believe more officers should carry guns.

The 22-year-old played for Yeovil in the 2012-13 season and briefly last term, but has been studying at Rider University for the past four years.
"I'm incredibly excited to be back playing for my hometown club who I have supported my whole life," she said.
"To have the opportunity to play for them in the top division is amazing."
Howard graduated two weeks ago with a major in history and a minor in French.
The UK unemployment rate has fallen to its lowest rate in more than a decade but wage growth has slowed.
[NEXT_CONCEPT]
Britain's most senior police chief has called for "less suspicion and more trust" in officers who carry guns, in his final speech before retirement.
[NEXT_CONCEPT]
Women's Super League One side Yeovil Town Ladies have re-signed goalkeeper Beth Howard after she graduated from university in the United States.