Despite producing an impressive team display at St. Jakob-Park in Sunday evening's second leg, the visitors could not find the goal required to take the tie to extra time as it finished goalless in Basel.
A controversial Ricardo Rodriguez penalty in the first leg ultimately proved the difference between the two sides, leaving a bitter taste in O'Neill's mouth as Norn Iron's wait for a fourth World Cup finals appearance goes on.
"We've gone out to a poor decision in the first leg," he told Sky Sports News. "Our performance and reaction tonight was fabulous. We were the better team. We had to ride our luck at times but we were pushing until the 94th minute. The players are devastated but I couldn't be any prouder.
"The first leg is difficult as it's tactical and cagey but tonight you're fighting for your dream and your life and the emotional side overrides tactics. Tonight we played with raw courage. We'll have some regrets. If it wasn't for the penalty we'd still be playing extra time now. They players will be remembered for the performance tonight.
"It's a very sore way to lose. We deserve to still be playing in extra time. We have to move on. We brought on young Jordan Jones, who is at Kilmarnock, and they're bringing on a £20m player. To take a team like Switzerland to the final minute over two legs is an amazing achievement. We don't always have top level players but we have top level characters."
O'Neill has yet to decide whether he will stay on as Northern Ireland manager, with a number of English clubs reportedly waiting to pounce should he step aside.