The Toffees, who had won at Manchester United in midweek, dominated for large spells at the Emirates Stadium, but fell behind to a Mesut Ozil goal on 80 minutes.
However, Martinez's well-drilled side produced an instant response as substitute Gerard Deulofeu, on loan from Barcelona, equalised.
Arsenal almost snatched a dramatic victory in stoppage-time when Olivier Giroud's 25-yard effort struck the woodwork, but Everton deserved to take something back to Merseyside.
Barkley was at the heart of Everton's forward momentum, and was involved in lifting his side for their deserved equaliser.
The 20-year-old has already been drafted into Roy Hodgson's plans for the World Cup, earning three full international caps.
Martinez, though, warned against expecting too much too soon from the Liverpool-born Everton academy graduate, whose talents have been compared to that of Paul Gascoigne.
He said: "I have never seen an English player with that sort of mentality. Ross has incredible potential, we need to give him time and accept he is not ready yet, that he will take time, but he has absolutely everything to mark a real era in our (English) football.
"Roy Hodgson needs to decide what role he wants for Ross, if any, we have to make sure we give him at least another 50 starts to see his evolution.
"Any player with talent needs to develop an understanding to know how to be effective, as a country we need to be a little bit more protective about our youngsters, because if we look after him properly he can give us something unique in English football and to the nation.
"He is developing an incredible awareness around him and technically he is someone who you could compare against any other nations, the young Brazilians, Dutch or Spanish players, so he has that gift of talent.
"As long as we give him time and that he can have development in front of him, he will be fine."
Everton had come into Sunday's match looking to extend a seven-match unbeaten Premier League run and follow on from their win over United.
"I was extremely pleased with the performance, in the first half we were magnificent and controlled the game, but if you play that well, you need to reflect it in the scoreline and that is our next step," Martinez said.
"We then showed incredible character after they scored, because it would have been easy to feel sorry for ourselves and the goal was a reward for our performance.
"We showed we have gotten over that mental block in going away to those teams who are going to be challenging for the title and are being ourselves, which is a massive step towards where we are trying to get."
Following other results on Saturday, which saw Chelsea lose and Manchester City draw, Arsenal were looking to put clear daylight between themselves and the chasing pack.
While it was not a maximum point's haul, Gunners boss Arsene Wenger was nevertheless pleased to have at least maintained momentum ahead of next week's Champions League trip to Napoli and then the showdown at Manchester City next weekend.
He said: "We played against a very good side, in the end we are a bit disappointed to get caught back with a 1-0 result, but overall it's still a point."
The Gunners boss was, however, somewhat frustrated by Everton's physical approach.
He continued: "When we got out of the pressure and got into the passing, they stopped us every single time with a foul. To just make a foul without being punished by the referee is a huge advantage because after that you play 10 against 10 in one half again."
Wenger hopes his side can maintain their slender advantage, with the visit of Chelsea set for December 23.
"As long as we are five points ahead, you cannot rule ourselves out even with the best desire, so let's keep that cushion," he said.
"It shows you that it will be a tough Premier League and the consistency in the end will be rewarded.
"The consistency for us is down to us to keep this attitude until the end of the season."
Everton signed Deulofeu, 19, on a season-long loan from Barcelona and could look to extend his stay on Merseyside.
"He has come a long way," Martinez said in his post-match press conference.
"When he arrived in pre-season, I thought it would take him to around February to get used to his role as a team player and the physicality of the Premier League.
"He has surprised me and everyone that he is ready now and wants to help the team. He is incredibly arrogant in the right way, has belief in himself and his finish against Arsenal was exactly what he deserved."
Martinez added: "We have an agreement that it could be extended an extra year as long as the parent club do not need the player."
Source: PA
Source: PA