A total of 1,366,026 passengers passed through its doors last month, up 5.8% on the same four-week period last year.
Of that total, more than 902,000 were international travellers and over 463,000 were passengers on domestic flights.
The figures come after the airport broke the 1.5 million mark for passenger numbers for the first time in July.
Chief executive Gordon Dewar said it was “another impressive month” for the airport, which has welcomed around 4.5 million people over the past three months.
“We have a fantastic range of destinations and airlines for passengers looking for that beach holiday or city break, and we’re always looking to add more, such as this month’s fantastic addition in Dubai with Emirates,” he said.
But he added that Scotland is recording growth “well below” the European average.
“We’re also seeing airlines announcing reductions and cuts to services due to us having the highest aviation tax in the world, something which is holding back the industry’s true potential,” Mr Dewar added.
“With the further challenge of an uncertain Brexit looming, the promised APD (air passenger duty) reduction needs to be delivered to avoid a stall in aviation and tourism growth.”
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