This was the fifth quarter in a row of growth, according to the national statistics office which published the figures.
It will publish final economic growth data for the period at the end of November.
The quarterly growth figures marked a slight slowing from growth of 0.6 percent in the second quarter.
Spain is climbing out of a deep recession triggered by the financial crisis, which led to the bursting of a big property bubble, and pushing the unemployment rate up to a record high of 26.9 percent during the first quarter of 2013.
The latest unemployment data showed that rate edged below 24.0 percent in the third quarter for the first time since 2011. But at 23.67 percent, it remains one of the highest in the industrialised world.
The country, with eurozone's fourth-biggest economy, clawed out of its second recession in five years during late last year, having applied radical reforms to strengthen public finances, the banking system and the labour market.