The Toffees are now 10 games unbeaten in the Premier League and only sit two points off the top of the table, with home games against bottom-club Sunderland and ninth-placed Southampton to come before the end of 2013.
Sunday's win at the Liberty Stadium marked Martinez's first against the club he both played for and managed in the 2000s.
"Huge credit to the team, it's such a pleasing win because it was a difficult game," the Everton manager said.
"We showed the desire and discipline to win the game and this is a very satisfying day."
On Barkley, who he has previously compared to Paul Gascoigne, Martinez added: "The goal from Ross gave us something to be proud of and something I want to savour.
"We have a few youngsters. Ross, John Stones, James McCarthy, Romelu Lukaku, Seamus Coleman... We have a real nice blend and what pleases me is that in the group, everyone is prepared to work hard for the team.
"Ross is growing game by game and is a diamond of English football. The free-kick was an example of what he can do."
Later, Martinez added: "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."
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.