The small uptick from the 0.5% rate recorded in June was mainly down to rising prices for fuel, alcoholic drinks and hotel rooms, according to the Office for National Statistics (ONS).
These upward pressures were partially offset by falls in social housing rent, and falling prices for certain games and toys.
The rate is still some way off the Bank of England’s official target of 2%.
This is the first piece of hard economic data since the EU referendum result on 24 June.
However, Ben Brettell, senior economist at Hargreaves Lansdown, said it does not paint a clear picture of the post-Brexit vote landscape.
He said: “The ONS collects data in the middle of each month, so the prices were collected just two or three weeks after the vote.
“It’s almost certain that the weaker pound will cause inflation to rise more sharply in the coming months, but the effect of sterling’s depreciation will take time to feed through fully into the figures as businesses gradually adjust to the new environment.”
Forecasts suggest CPI inflation could ultimately reach 3%.
However, Brettell said this will be a temporary factor: “Assuming sterling remains weak, the effect will fall out of the year-on-year calculation in the second half of next year.”
The Bank of England is widely expected to leave rates on hold at its next meeting in September, though swap markets are pricing in around a 33% chance of a cut to zero by the end of the year.
The increase in CPI spells more bad news for savers, already struggling with record low interest rates, as inflation creeps towards the average return on savings accounts.