From what I know about the alarm, that seems to be a common complaint and I'm not sure you can change the behavior (I can't on my S3). It might be worth checking out other alarm clock apps, or setting a tone that's so annoying you need to turn it off immediately rather than letting it go off for a minute
When alarm is ringing, you have two controls: red one to shut it down completely and yellow one to snooze it for amount of time specified in settings. Stock alarm app has an intelligent alarm option too, just set your repeat interval and alarm sound and you're good to go.