http://developer.force.com/cookbook/recipe/running-case-assignment-rules-from-apex


//Fetching the assignment rules on case
AssignmentRule AR = new AssignmentRule();
AR = [select id from AssignmentRule where SobjectType = 'Case' and Active = true limit 1];

//Creating the DMLOptions for "Assign using active assignment rules" checkbox
Database.DMLOptions dmlOpts = new Database.DMLOptions();
dmlOpts.assignmentRuleHeader.assignmentRuleId= AR.id;

Case newCase = new Case(Status = 'New') ;
//Setting the DMLOption on Case instance
newCase.setOptions(dmlOpts);
insert newCase ;



Use below trigger to fire the assignment rule on case creation via API:

trigger CaseAssignment on Case (after insert) {
    //Variable decleration
    List<Case> caseList = new List<Case>();
    User integrationUserObj = new User();
   
    if(Trigger.isAfter && Trigger.isInsert){
        //Fetching the integration user details
        integrationUserObj = [SELECT Id, Name FROM User where Name = 'Integration User' LIMIT 1];
           
        for (Case caseObj : Trigger.new) {
            if (caseObj.OwnerId  == integrationUserObj.Id) {
                caseList.add(new Case(id = caseObj.id));
            }
        }
   
        Database.DMLOptions dmo = new Database.DMLOptions();
        dmo.assignmentRuleHeader.useDefaultRule = true;
        Database.update(caseList, dmo);
    }
}