The midfielder produced a contender for goal of the season to open the scoring as he picked up the ball deep in his own half before charging at the home defence and smashing the ball home from inside the area.
The 20-year-old is among several candidates vying for a place in Roy Hodgson's England squad for the World Cup this summer and the Spaniard is adamant that he would be ready if called upon.
"That's Ross Barkley: not many players can score that sort of goal," Martinez told the press.
|4/1||Everton are 4/1 with Bet365 to finish in the Premier League top four|
"If the national team manager believes he needs the role of Ross, he'll be ready. I know that Ross Barkley will be a phenomenal footballer for England and for English football but it has to be managed by the manager of the England setup and we have to respect that."
Their victory put the Toffees six points off fourth-placed Arsenal, thanks to the Gunners' 2-2 draw against Swansea City, with a game in hand and a home clash with Arsene Wenger's side still to come and Martinez remains hopeful of "springing a surprise" with Champions League qualification.
"It's not a matter of believing or not," he opined. "We always felt we were going to finish strong. We want to get as many points as we can.
"There's going to be a lot of twists and turns. I just feel that, if we can get anywhere near 71, 72 points and that gets us in the Champions League, it would be an outstanding achievement.
"You never know but I do feel that the league can still spring a few surprises in every area from now until the end of the season."
John Carver took charge of affairs for the home side in the absence of Alan Pardew, as the manager completed his stadium ban, and he could not resist commenting on Barkley's impressive display.
"When you've got the ability like he does and when you can finish the way he finished it then you can't do anything about it," said the Magpies' assistant boss.
"They were so clinical in their finishing; we weren't again. We came up against a very good side. We matched them that first half and it was only a bit of sheer genius that swung it."