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.
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."