Barkley opened the scoring for Everton in the 3-0 win at St James' Park, finding the net with a magnificent solo goal that saw him race from inside his own half before cutting past several Newcastle defenders and powering a fierce strike into the top corner.
Romelu Lukaku doubled Everton's lead early in the second half before Leon Osman rounded off a superb team performance with a fine third.
But the plaudits went to Barkley, who Martinez was quick to praise both for his goal and for an excellent all-round display.
"Not many players can score that sort of goal," Martinez said. "I think that Ross Barkley is technically as good as you are going to get, but he's got that physicality to go with it. So that goal typifies what Ross Barkley is.
"The goal, you can admire it and see the style, but what pleased me more from that point for a young player, instead of losing the concentration about his role, he was perfect, he worked extremely hard for the team and it became a real mature performance from his point of view."
Martinez was delighted with Everton's showing at both ends of the pitch, and is confident that the Merseyside club can finish the campaign strongly following a victory that moves them within six points of fourth-placed Arsenal.
"We were very powerful in both boxes," he added. "We defended well when we had to and to keep a clean sheat against opposition like Newcastle, it shows you the focus, the concentration and intensity that we had.
"Then every time we went forward I felt that we had a real threat. To score three goals in the manner that we did is probably the difference between playing well and not getting a positive result and playing well and getting the three points.
"Today the focus, intensity and the momentum of that performance allows me to believe that these players are going to finish really, really strong in the next few games."
