The Italians, four-time winners and champions as recently as 2006, took an early lead in Zenica, but after going down to 10 men, they were hauled back by Bosnia before penalties sent Gattuso's side packing.
After suffering defeat at the playoff stage for the last two World Cups, another absence will be hard to take for all involved in Italian football.
"Today the boys didn't deserve a beating like this," a teary-eyed Gattuso told RAI.
"We were left with 10 players, we had three decent chances and they barely troubled us. I'm sorry, this is football and I'm proud of my boys.
"It hurts, because we needed it for us, for all of Italy and for our movement. A blow that's difficult to digest."
Gattuso and Italy were incensed when a foul by Tarik Muharemovic on the edge of the area brought just a yellow card early in extra time, but the manager did not want to talk about refereeing decisions.
"I don't want to talk about anything, but today it's unfair," he said.
"I have been in the world of football for years, sometimes I have rejoiced and today I take a beating. It's hard to digest, they surprised me too for the heart they put into it.
"We're talking for the umpteenth time that we don't go to the World Cup. I apologise if I didn't make it, but the boys impressed me today."
'My future is not important'
Gattuso, who replaced Luciano Spalletti early in the qualification campaign after defeat to Norway, was asked about his future.
"Today talking about my future is not important, it was important to go to the World Cup," he replied.
"We hold onto this performance, but it hurts and I'm sorry."
The head of the Italian Football Federation (FIGC) Gabriele Gravina told reporters that he had asked Gattuso to stay on.
"The mood is quite evident, especially for how this result has matured," he said.
"Allow me to congratulate the boys, in recent months they have had incredible growth, many of you have not been able to appreciate the climate and atmosphere in recent months.
"I want to congratulate Rino Gattuso. He is a great coach, I asked him to stay at the helm with these boys. There is little to comment, the coach called them heroic. Then, I understand there are several evaluations to be made."
Gravina is himself under extreme pressure at the head of Italian football given their repeated failures.
"For the political part there is a place to make the evaluations, namely the Federal Council," he said.
"I have already decided to call them up for next week, there will be evaluations made internally.
"I understand the request for resignation given the situation, but there is a suitable place to make the evaluations."