Pilots on board began the landing process shortly after the fire was reported as the plane flew over the English channel.
They landed the aircraft in Brest, where fire tenders and police were waiting on the runway.
Passengers were taken off the plane shortly after.
According to a flight-tracking platform, the plane made a rapid descent over the course of seven minutes, dropping from 41,000 feet to 6,725 feet.
Passengers from the plane were transferred to another aircraft at Brest to continue their journey to Portugal.
All customers have also been provided with a hotel stay overnight before continuing their journey to Dublin in the morning.