"What we saw was tremendous resistance by the permanent members of the security council to keep their exclusive power," Richard Dicker from Human Rights Watch said.
"And on the other side the staunch insistence of states to preserve the principle of independence of the court from interference."
The agreement also includes a review clause, delaying its entry into force until ICC member states grant formal approval after January 2017.
The crime of aggression is broadly defined as the use of force that manifestly breaches the UN charter and includes an invasion, a bombardment, blockade or a country allowing another state to use its territory to attack a third nation.
Ichiro Komatsu, Japan's head of delegation, criticised the resolution, saying that there were doubts about its legality and that it undermined the credibility of the Rome Statute under which the ICC was set up.
Komatsu said he was "deeply concerned" about the policy direction of the resolution.
"A government of a state party surrounded by non-state parties will have difficulty selling to its parliament an amendment which unjustifiably solidifies blanket and automatic impunity of nationals of non-state parties," he said.
But Christian Wenaweser, president of the Assembly of States Parties that overseas the ICC, said he did not share the same concerns as Japan.
"This is what I came in here to achieve and that is what we've been able to achieve," Wenaweser said of the agreement, adding that he was confident of the continued support of Japan, the court's largest financial backer.
"Two weeks ago I didn't necessarily expect that we would be able to find a solution that dealt with all aspects of the crime of aggression, so yes it is a bit more than I initially hoped for," he said.
Several other delegates said the deal had come about only because certain countries were prepared to give way on some sticking points after days of intense negotiations.
The issue has divided countries over the role the UN Security Council should play.
Non-governmental organisations said that granting the security council sole power to authorise an investigation would have hit the court's independence.
Latin American and African nationshave been wary of yielding authority to a world body dominated by the five permanent Security Council members - Britain, the US, China, Russia and France.
"The agreement may pave the way for the court to act on aggression without Security Council control," Elizabeth Evanson, a lawyer in Human Rights Watch's international justice programme, said.
"This required ICC members to resist pressure from permanent members of the Security Council to keep that control."