The Toffees roared to their first away win of the calendar year as Barkley, Romelu Lukaku and Leon Osman all found the net.
England midfielder Barkley provided further evidence to support his claim for a World Cup place with an exquisite solo effort to break the deadlock midway through the first half.
The 20-year-old picked up possession 70 yards out and charged into Newcastle territory - leaving a host of defenders in his wake - before firing beyond Tim Krul, and Martinez applauded the efforts of his highly-rated playmaker.
"I think Ross has got that sensational talent," Martinez told Sky Sports.
"You've seen that physicality; you don't get many players with that technical ability that can come with that physical element.
"He has both, to run across the whole pitch and score, it was a real joy to watch."
Everton doubled their lead shortly after the break when Gerard Deulofeu teed up Lukaku to crash home the visitors' second, before captain Osman completed the haul three minutes from time.
Not only did Martinez's side record a first win on the road since 2013, it was their first season double over Newcastle since 1989 and the Spaniard was delighted with Everton's complete display.
"We knew it was going to be difficult, you know you have to be perfect to keep a clean sheet at St James' Park," he added.
"I thought our performance was complete in terms of our work rate and counter-attack. I thought we controlled the game for long periods of time.
"It was an away performance as good as it gets and perfect timing, the understanding of what was needed today was excellent and the players who came in put in a real shift."
Newcastle made the brighter start on Tyneside as Luuk de Jong, Yoan Gouffran and Papiss Cisse all squandered chances, before Lukaku was denied by Krul, leaving Martinez to question whether his side would find their clinical edge.
"I think in previous away games we performed well but we haven't been good enough in front of goal," he admitted.
"We had a great chance when Romelu Lukaku got in a great position but Tim Krul produced the save of the match.
"At that time it was a real question of how we were going to react. I thought the way we reacted with that desire, wanting to score goals."
Victory elevates Everton to fifth in the Premier League standings, six points behind Arsenal with a game in hand, but the manager remains focused on his club's development.
"I think it's fair to say that this year's competition is as good as we've seen for many years," Martinez claimed.
"It is as good as you're going to get. When you get that sort of competition, you have to look at yourself, you can't afford to look elsewhere.
"I don't think we look at Arsenal, we must internally as a group work towards a points tally. When you see such an unselfish performance like today, that's what the fans are proud of."