Italy coach Cesare Prandelli admitted two missed chances by Mario Balotelli had proved costly as Costa Rica triumphed 1-0 to clinch their place in the last 16.
Bryan Ruiz's header secured a 1-0 win for the Ticos, who will qualify from Group D even if they lose to England on Tuesday.
He said: "Mario played well in the first half. He had two chances and if he had scored then the game would have been different.
"We came into this game with too slow a pace, and Costa Rica were well-organised and so our approach was not of the best."
Asked if he had a message for England, who had been relying on an Italian victory, Prandelli added: "I am terribly sorry – not just for the English but also for us as we wanted to win.
"Now we will have to play well against Uruguay in order to qualify. We knew this was a particularly challenging and difficult group and now we have to try to recover our energy."
Jorge Luis Pinto, Costa Rica's Colombian coach, said he was now targeting a third win in the group – against England.
He said: "Of course we want to win the next match now and I must ask my players to keep calm and play just as well.
"We knew that we could make history for Costa Rica and we have done that. That's what I told my players – let's change history.
"It was a beautiful match and the people of Costa Rica deserve this. They have supported the team and me and this is for them."
Costa Rica showed their 3-1 win over Uruguay was no fluke as they took the game to Italy in impressive style.
Celso Borges headed over from a corner early on, but it was Italy's Mario Balotelli who had the best of the chances in the first half-hour.
First, Andrea Pirlo's glorious ball over the top dropped perfectly for Balotelli, who controlled it with one touch and then lofted it over the on-rushing keeper Keylor Navas, but it bounced well wide.
Two minutes later and Balotelli was in again. This time he went for power but shot too close to Navas, and the Costa Rica keeper was able to claim.
With half-time looming, Costa Rica were denied a blatant penalty.
Joel Campbell intercepted a pass, surged into the penalty box and was knocked over from behind by Giorgio Chiellini but Chilean referee Enrique Osses inexplicably waved play on.
Costa Rica did not let that put them off their stride however and Junior Diaz's exceptional cross from the left saw Fulham striker Ruiz thump a far-post header in off the crossbar.
Italy upped their game after the break. Balotelli claimed a penalty of his own under a challenge from Oscar Duarte, but the referee once again turned away.
Italy mounted a series of attacks but Navas was having a blinder in the Costa Rica goal.
Randall Brenes was a yard away from scoring a sensational second in injury time but it mattered not – Costa Rica were through and England's fate was sealed.