Flights were delayed and cancelled on Tuesday evening after a high-voltage transmission pole was damaged.
Some terminals were left without overhead lighting before the power was restored several hours later.
The airport's chief executive Kevin Brown says a review will be undertaken into why some back-up generators failed.
"The back-up generation system for critical safety systems such as the runway lighting worked as intended, meaning that aircraft could continue to land safely throughout the event," Mr Brown said in a statement on Wednesday.
"Other parts of the back-up generation system that provide power to the terminals did not work as intended. We need to understand why that happened."
Mr Brown apologised to passengers for the inconvenience caused by the outage as the airport dealt with a "unique and challenging weather event".
He said the review would examine whether the airport could work more closely with Western Power on the transition to back-up power.
"We face a busy start this morning as we work with our airline partners to clear the backlog of delayed flights and we again thank passengers for their patience," he said.
Around 8100 homes and businesses remained without power on Wednesday morning due to storm-related outages.
Most were in the Perth metropolitan area, with others in Western Australia's southwest.
"We understand the frustration and inconvenience this has caused, particularly for those who were without power overnight," Western Power said.
"Our crews and network operations centre worked through the night in difficult circumstances to make hazards safe and restore power where they could, with around 1000 homes restored during the night."
A spokesperson said the damage to infrastructure overnight was less severe than on Tuesday when a peak of about 35,000 customers experienced disruptions.
Emergency services have answered more than 440 calls for help since Tuesday afternoon, with fallen trees bringing down power lines and damaging fences.
The wild weather is expected to continue in Perth with the weather bureau forecasting damaging winds with peak gusts exceeding 100km/h.
Conditions are expected to ease later on Wednesday.
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