Arena's side needed just a draw from their final qualifying match against Trinidad & Tobago on Tuesday night in order to qualify but instead suffered a shock 2-1 defeat, leading the former Los Angeles Galaxy coach to tender his resignation.
"It is the greatest privilege for any coach to manage their country's national team, and as I leave that role today I am honoured and grateful to have had that opportunity twice in my career," he said in a statement.
"When I took the job last November, I knew there was a great challenge ahead, probably more than most people could appreciate. Everyone involved in the program gave everything they had for the last 11 months and, in the end, we came up short. No excuses. We didn't get the job done, and I accept responsibility.
"Obviously the biggest disappointment is for our fans. As a person involved in the sport for more than 40 years, to see how support for soccer in the United States has grown is incredibly gratifying.
"While this is a difficult time, I maintain a fierce belief that we are heading in the right direction. I believe in the American player and the American coach, and with our combined efforts the future remains bright.
"I don't know what the future holds for me, but I can say this from the bottom of my heart: from the high of reaching the quarterfinals of the 2002 World Cup to the low of a few days ago; I have appreciated every minute of being a part of this program."