With the scores at 1-1 in the last moments of extra time, Dominic Adiyiah's header was stopped on the goal-line by the hand of La Celeste forward Luiz Suarez, who was consequently sent off. However, Asamoah Gyan missed the resulting spot-kick before the Black Stars lost 4-2 in the shoot-out.
"There are no words to express what we feel," said Rajevac. "But I congratulate Uruguay tonight - they were the lucky ones. All I can say is that this is football. At the end we had an historic opportunity to reach the semi-final - we had a penalty.
"The opponents had the psychological advantage in the shoot-out. Everything happened so fast. We really had big support and it would have been a fairytale if it ended well for us, but I would like to congratulate Uruguay on reaching the semi-finals.
"I'm very proud. We managed to achieve a great result and the whole of Africa supported us. We didn't deserve to lose in such a way and it's difficult to talk about it at the moment. We had bad luck that's all I can say."
Uruguay, meanwhile, were left celebrating their first FIFA World Cup semi-final place in 40 years - they will play the Netherlands in Cape Town on Tuesday - and the result sparked joyous scenes in the country.
"I hope the party goes on for three or four days!" said Tabarez. "I am the coach of the team, I'm a pro, and even so I lack the necessary calm to carry out an objective analysis of what happened.
"We achieved our goal but we didn't play as we wanted to. Our rivals were extremely tough. We were lucky.
"Luck is important and I really cannot say much more. Those who believe in fate or destiny might be able to explain it. I don't believe in that."