Jim Reid, 75, and David Irvine, 35, joked that they passed the time cursing each other after getting lost in fog on an early-morning lobster-fishing trip in their 16ft creel boat on Tuesday.
The experienced fishermen left Gourdon harbour at 4am on Tuesday but became disorientated when their compass stopped working. They drifted 46 miles out to sea, where they vanished into thick fog.
The alarm was raised when they failed to return at noon on Tuesday.
Coastguard search teams were scrambled along with two rescue helicopters and three lifeboat crews.
The pair were reunited with their families after a passing fishing vessel spotted them and rescued them at about 7.30am yesterday.
Mr Irvine said: "The compass wasn't showing us where to go and because of the fog on the first night we were completely lost.
"We had no idea where we were. We followed the compass but it just wasn't the right direction. We couldn't see the land. We were trusting the compass completely."
The pair ended up miles from the shore, trying to conserve fuel while surviving on a litre-and-a- half of water, a flask of tea and two biscuits.
Mr Reid said they saw several big ships as they drifted out to sea but they were too far away to summon.
They had no idea of the huge search taking place for them closer to shore.
He said: "I had two rockets which I set off on the first night. What was worrying me more was what was happening back home. We never gave up hope."
Mr Reid joked he came back from his ordeal two stones lighter, and that they passed the time adrift in the North Sea by "cursing each other".
He said he was elated when the trawler found them and added: "I just about walked to it. There was a comedy about it. They were coming towards us and I was standing on my boat with a red ball, indicating.
"I'm waving it to them and they just went straight past - you should have heard my language. But they came back around. It was just the best thing in the world."
Mr Irvine added: "It was a mix of relief and shock. We had seen so many ships that were far away that we couldn't reach."
Mr Reid said that despite 60 years experience of fishing he had now learned the lesson that he had always to be prepared.
Relieved family members gathered at the harbour in Montrose waiting for the rescued fishermen to return home.
Mr Irvine's mother, Marion Irvine, 53, said: "I thought I was dreaming. I don't know what's happened to them on the water and there was always hope they would be found.
"I have been tearful for the past few days and this is just unreal."
Mr Reid's sister Dorothy Milne said the family suffered a tragedy at sea when her nephew Johnny, 19, the son of her oldest brother James, died after falling from a creel boat 25 years ago.
Ms Milne said they never thought there would be a happy outcome to the latest incident. She said: "They phoned earlier this morning to say they'd been found. I just got a call to say they had found them. I said, 'Well, there must be a God'. It must have been awful for them."
Their rescuers towed their boat back but it sank before it reached harbour.