Others who had landed were having trouble collecting the luggage, with disgruntled travellers complaining of "massive crowds".
Some people appeared to be locked out of the airport altogether.
The outage was caused by wild weather which lashed Perth on Tuesday. Winds with speeds of up to 70km/h battered the west coast on Tuesday, and the Bureau of Meteorology expects the turbulent weather to continue until Thursday.
It is not known how long the damage will take to fix.
"Perth Airport will be delaying all outbound services scheduled to depart before 7.30pm," the airport tweeted.
"All inbound flights currently in the air will be able to arrive safely. All scheduled flights into Perth which have not yet departed will be delayed until further notice.
"Passengers should expect delays/cancellations. The safety of everyone who works in or is travelling through our airport remains our highest priority. We apologise to passengers and customers for any inconvenience."
At 2.34 am the airport published another Tweet reporting that power had been restored.
"We recommend passengers check with their airline directly regarding any flight status updates," the tweet read, before asking passengers continue to show patience towards the airport team and airline crew.
One passenger responded, saying they had been waiting for nine hours without updates and had their flight cancelled.
— news.com.au with additional reporting