You may be inadvertently saving the event to the wrong calendar. Often email accounts, such as Gmail automatically enable a calendar when being set up. If the event shows up only on your iPhone, and nowhere else, chances are your default calendar is set to something that you are not used to checking.
When you create an event, scroll down to where you can see the menu item "Calendar". Tap that, and choose one of your iCloud calendars. If the event then shows up on iCloud, and your computer, you have chosen the correct calendar.
If this is case, you will probably want to change your default calendar by opening your Settings.app, scrolling down and selecting "Mail, Contacts, and Calendar". At the bottom of this screen is a section titled "Calendars". Change the "Default Calendar" to reflect the one that worked correctly.