The men were forced to abandon ship and managed to get from a lifeboat into a nearby vessel which had come to their aid.
Shortly afterwards RNLI rescuers transferred two of the casualties to their boat for medical attention while the third was airlifted to hospital.
The incident occurred about 30 miles of the coast on Thursday afternoon with an emergency distress call alerting the RNLI at about 3.45pm.
The cause of the fire is not clear. The 14 metre unnamed trawler eventually sank despite the efforts of a tug boat with fire-fighting capability to extinguish the flames.
“Thankfully, all three fishermen were rescued this evening and we would like to wish them all a speedy recovery following what must have been a frightening experience,” said Arklow RNLI coxswain Ned Dillon, who praised the crew members for taking the decision to abandon ship.
“It is always sad to see a fishing vessel sink but we are delighted that all three fishermen are safe and recovering from their ordeal this evening.”
Mr Dillon also thanked a number of vessels in the area who had responded to the situation as well as Wicklow RNLI and Irish Coast Guard responders.
The Wicklow and Arklow lifeboats were supported by the Irish Coast Guard helicopter Rescue 117 from Waterford.