Seamus Coleman had given the visitors a 66th-minute lead with a ferocious strike, but they were pegged back when Bryan Oviedo put Dwight Tiendalli's off-target effort into his own net.
Everton were not to be denied, though, as Barkley stepped up with a free-kick which nicked the underside of the crossbar on its way in.
The goal, and another fine individual performance, will ensure the 20-year-old will again be in the headlines after a string of excellent recent displays, and it is hard to see how Roy Hodgson can leave him out of England's World Cup squad.
The win moves Everton up to fourth in the Barclays Premier League table, and Martinez was left to heap more praise on his midfield starlet.
"Ross is growing game by game. He is showing different aspects of his play, he is a quite incredible diamond of English football," said the Everton manager.
"The winning goal, technically, is as good as it gets. A lot has been said about Ross, but it's always an understatement.
"He's an incredible listener and young man with a lot of potential. He always comes up with something different in a game to show he can develop.
"I'm proud of seeing him grow.
"You could see he was fighting with Romelu because Romelu really practices his free-kicks.
"But Ross is a player who gets fed by feelings. He's a natural talent who can play in any position.
"We didn't know he could take free-kicks, but it's good to know from now on."
Everton thoroughly merited their success, which tees up back-to-back home games against Sunderland and Southampton.
Although the Merseysiders were not at their best, they still created the majority of the significant chances.
And Martinez felt the win at one of his former clubs was as satisfying as any win during a season which has also included successes against Chelsea and Manchester United.
"We had three difficult away games. At Old Trafford, the Emirates and the Liberty and this was the biggest test," he said.
"It's easy to keep your concentration levels when you go to the other two, but unless you are 100 per cent when you haven't got the ball, this Swansea side will punish you.
"I couldn't be happier. We weren't at our best, but we kept Swansea quiet. We conceded a soft goal by our standards, but we had the desire and our determination to want to win the game was pleasing.
"It's the best satisfaction of the season."
Martinez was also confident the problem which saw Steven Pienaar limp off late in the game will not keep the South African out of the Boxing Day meeting with Sunderland.
Swansea manager Michael Laudrup felt his side should have got something from the game.
"I think we deserved more," he said. "The first half was tight, there were not many chances but they came out second half and went in front with a strike from distance.
"We fought back as we really wanted something out of this game but the thing that is most disappointing is I thought in the last 10 minutes they were happy to take a point.
"We were still pushing to get a winner and then they got it from a free-kick.
"But we have to accept it."
Laudrup also had praise for Barkley, saying: "He is a great talent, he is a great runner with the ball, he can play in tight areas, he is definitely a big talent.
"He already has a lot of talents for someone so young."
Laudrup confirmed Swansea are set to make a decision on whether Michu undergoes surgery on an ankle injury.
"We will know more tomorrow," he said. "He may have a chance for Boxing Day but we will see, we will have a final opinion from an expert and then decide.
"If tomorrow the information suggests he needs an operation he will have surgery in the very near future.
"Obviously we are not happy he could be out for several weeks, but the most disappointed person is Michu.
"If it is bad news tomorrow, added to what happened to Nathan Dyer last week, we will have to think about what to do in January."
Source : PA
Source: PA