The map upset the Vietnamese Football Federation because it did not show the disputed Spratly and Paracel Islands as belonging to Vietnam.
In a statement, Arsenal said that it regretted any offence caused.
The Spratly Islands are also claimed by China, while the Paracel Islands are claimed by six countries in the region.
"We have a great working relationship with the Vietnamese Football Federation, who brought to our attention that the map of Vietnam in one of our promotional videos for our forthcoming visit did not include the Hoang Sa (Paracel) and Truong Sa (Spratly) Islands," the Arsenal statement said.
"Arsenal Football Club does not sympathise with any political issue and consequently there has been absolutely no intention to cause offence to any country or political cause."
A spokesman told the BBC that all future promotional videos made by the club would not contain maps. He said Arsenal was now looking forward to the tour of Vietnam going ahead next month.
Vietnam is especially sensitive about the disputed islands. In November it refused to stamp new Chinese passports bearing a map that laid claim to them.
The Arsenal spokesman said it was impossible to stop the circulation of the video promoting the match - the first such game to be played by a Premier League club in Vietnam - because it was widely available on YouTube and impossible to remove.
The 17 July match against the Vietnamese national team was nearly cancelled earlier this month following another row - this time over the cost of renting of Hanoi's My Dinh National Stadium.