Seven protesters from the campaign group Plane Stupid were arrested by Grampian Police at around 8.20am.
Two who remained on the roof of the main terminal building were detained later.
Flights to and from the airport were temporarily suspended when the protest began at around 3.15am but later resumed at the time of the first arrests.
Some knock-on disruption to flights is expected during the course of the day.
Members of Plane Stupid, who were dressed in the style of property tycoon Donald Trump, hung a banner across the front of the terminal building saying 'Nae Trump games with climate change'.
Trump is backing the expansion of the airport following approval being granted for his £1bn golf project in Aberdeenshire by the Scottish Government.
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Protester Tilly Gifford, 24, from Glasgow, said the demonstrators had set up a makeshift golf course within a fortified fence.
During the protest Ms Gifford told BBC Scotland: "Four of the protesters have their necks locked to the fortified fence.
"We are trying to close down the airport. We can measure its success in terms of stopping carbon emissions.
"We are here to highlight that aviation is the fastest growing source of carbon emissions."
Another protester, Dan Glass from Glasgow, said: "We're here to say the expansion cannot happen, and our generation won't let it happen."
The protesters are linked to those who caused disruption at Stansted Airport last December by occupying the runway.
A Grampian Police spokesperson said: "All nine people, five of whom are males and four of whom are females, have now been taken into police custody and the incident has been resolved peacefully.
"Grampian Police are currently working with the BAA to minimise the impact this incident has had.
"An investigation into the incident is currently ongoing at the moment and a report will be submitted to the procurator fiscal in due course. As such it would be inappropriate to comment further at this stage."
A spokesman for Aberdeen Airport said: "This action is dangerous and highly irresponsible, not least because Aberdeen is one of Europe's busiest commercial heliports and a major transport centre for the north of Scotland, used by tens of thousands of people every day.
"There is no justification for this irresponsible action, which is deliberately calculated to delay and inconvenience the travelling public."
Passengers have been asked to check the status of their flights.
(GK/BMcC)
