Minasyan's side picked up several impressive results in Group B, including wins in the Czech Republic and Denmark, crushing Morten Olsen's side 4-0.
They also ended the group phase with a creditable 2-2 draw against Italy in Naples, but just one win and four defeats from their home fixtures destroyed their chances of reaching next year's tournament in Brazil.
They ultimately finished second from bottom, above Malta, who gained their only points of the campaign with a shock win in Yerevan in June.
And Minasyan, who has been involved in the international set-up since 2006, has chosen to end his four-year reign over the national side, in the hope of securing a club job in Europe.
He said: "I prefer another continuation to my career. There are several offers, specifically from abroad, and I want to discuss them.
"I would like to thank Ruben Hayrapetyan, who entrusted me the duties of the national team head coach for the past four years. It is a great honour for every coach to be the manager of a national squad.
"I also would like to thank our football fans. I have always felt their support. I am confident that they will always be with our national team, our football.
"I also would like to thank the media representatives. I have always been honest with them.
"And most importantly, I want to thank our players. It has been a great pleasure for me to work with them. Together we passed along a road which, I believe, was useful. They have a great potential, they are professionals, and I am confident that they still will climb great heights."
