The 20-year-old lit up St James' Park, where a young Paul Gascoigne announced himself to the football world in such style, to invite comparisons with the star of England's 1990 World Cup finals campaign.
Barkley's contribution capped a fine display by the Toffees as they ran out 3-0 winners on Tyneside, and his manager was fulsome in his praise.
Martinez said: "To be fair, he has got such a variety of strengths. He can do things that only top players can do. I don't think he has got a like-for-like as a footballer.
"Sometimes I am looking at him and he does things that remind you of Michael Ballack; at times it's Paul Gascoigne.
"He has got a uniqueness about his play. Technically, he is as gifted as you are going to see, right and left, you wouldn't see a difference.
"But what shocks you is the physicality that goes with it. Then on top of that is the character. Nothing distracts him from his role.
"He is a really down to earth, modern footballer who all he wants to do is enjoy his football and score goals. He is a joy to work with."
Martinez has spent much of this season being asked about Barkley's readiness to be part of Roy Hodgson's squad in Brazil this summer.
While he refused to champion the youngster's cause on Tyneside, he insisted he would be ready if Hodgson came calling.
The Spaniard said: "All I can say is that Ross is ready to help the national team, and nothing will faze him.
"If the national team manager believes that he needs the role of Ross, he will be ready; if not, he will have a good summer and he needs to regenerate and recover after what has been his best and most demanding season as a professional.
"I know that Ross Barkley will be a phenomenal footballer for England and English football.
"But the timing has to be measured by the manager in the England set-up and we all have to respect that because every manager prepares a certain squad for different reasons.
"Ross is a proud Evertonian, he loves playing football and is a proud Englishman, and he would be very, very proud to represent England whenever he is called."
Barkley ran from his own half and carved his way through the home defence before blasting home his 22nd-minute opener, and Romelu Lukaku's 13th goal of the season extended the visitors' advantage eight minutes after the restart before Leon Osman made sure of the win three minutes from time.
Newcastle assistant boss John Carver, who stood in for manager Alan Pardew for the final time as the 52-year-old completed his three-match stadium ban, was well-placed to assess the comparison Barkley and Gascoigne having been on the playing staff at St James' as the precocious Geordie emerged from the ranks.
He said: "I think that's a lot of pressure to put on the lad, but I think he's a class act.
"I'm sure their manager will nurture him in the right way and he's got good people around him.
"He's got a bright future - we've just got to be careful that we don't build him up too soon.
"He's a Ross Barkley, not a Paul Gascoigne, but as long as they manage him right and we as a country manage him right, I think he's got a bright future."
Victory on Tyneside, coupled with Arsenal's 2-2 home draw with Swansea, saw Everton climb into fifth place, six points behind the Gunners with a game in hand ahead of the clash between the two at Goodison Park on April 6.