The electricity supply went out at around 1.30am but has since been restored, although it will take time to get systems up and running again.
Bobbi Hadgraft, who is travelling to watch England at Euro 2024, told BBC Radio Manchester she arrived at the airport at about 4am to see “enormous queues” outside Terminal 1.
“We were worried we would miss our flight,” she said, adding the power outage had affected display boards and scanners so staff had to direct passengers around the site.
Meanwhile, rail passengers suffered travel chaos on Sunday because of a signalling problem on busy routes.
South Western Railway urged people not to attempt to travel between Brockenhurst and Eastleigh or between Fareham and Southampton as lines were blocked.
A message to passengers said: “We are currently unable to operate a service between Brockenhurst and Eastleigh in both directions.
“We are also currently unable to operate a service between Fareham and Southampton Central.
“Train services running through these stations may be cancelled, delayed or revised. Disruption is expected until the end of the day.”