The British statistics office says there's been no obvious Brexit-effect
The UK's Consumer Price Index (CPI) measure of inflation rose to 0.6% in July as increased fuel costs drove up transport prices.
The CPI figure was up from 0.5% in June, the Office for National Statistics (ONS) said.
Economists had been predicting the CPI figure would remain unchanged, but the increased transport costs - including the cost of second-hand cars - contributed to the rise.
Mike Prestwood, head of prices at UK's ONS, said: "The Consumer Price Index has continued in July its recent slow upward trend since late 2015, with transport costs the biggest single factor this month.
"There is no obvious impact on today's consumer prices figures following the EU referendum result, though the Producer Prices Index (PPI) suggests the fall in the exchange rate is beginning to push up import price faced by manufacturers.
"These are the first sets of consumer and producer prices data collected since the referendum polling day."
The CPI measures changes in the cost of a basket of consumer goods and services typically purchased by households - including lemons, nail varnish and computer game downloads.
In contrast, the Retail Price Index (RPI) rose to 1.9% in July.
The RPI includes the basket of consumer goods, but also costs from running a household including changes to mortgage rates and council tax payments.
The ONS said the CPI was affected by a 1.6% rise in transport costs between June and July this year - compared with a 1.2% increase over the same period in 2015.
The cost of alcoholic drinks also increased by 0.5% month-on-month in July, compared with a fall a year ago.
But the price of housing, water, electricity, gas and other fuels - which were unchanged over the period - put downward pressure on the CPI.
The price of petrol climbed from 111p in June to 111.8p a litre in July, while the cost of diesel rose from 112.1p to 113p a litre in the same period.
It comes after the Bank of England cut interest rates to a historic low of 0.25% and delivered an emergency package of measures worth up to £170bn to stave off recession following the Brexit vote.
Official figures show the UK economy had grown in the run-up to the EU referendum, with gross domestic product (GDP) growing 0.6% in the second quarter.
But think-tank the National Institute of Economic and Social Research (NIESR) estimated that GDP fell by 0.2% month on month in July following the vote.