When your throat closes as described, you're actually pushing your voice from your throat and not from your diaphragm. Proper breathing is the key to doing it correctly, but there are other things you should do when singing.
Vocal warmup exercises are excellent to open up your throat, just relax, take a deep breath and warm up until you are ready to sing.
Relax... This is extremely important. Do not sing when feeling uncomfortable, as you'll probably force your voice too much. Drink some water or hot tea, sit down and focus, hum the melody you want to sing and try it again when you're ready.
Checking your throat in a mirror is an advice my singing teacher gave me which works wonders. Stand in front of a mirror and sing a melody you're very comfortable singing. Check your throat: doesn't move at all, right? your Adam's apple isn't moving and should not move, until you reach the highest notes you can actually sing. Try higher notes and check if your throat tightens up now, the lesser it moves, the better. If it doesn't work, try pushing your voice back to the edge of throat when singing, the air should come out directly from the lower end of your throat.
Roleplay. Try sounding like Santa Clause or an Old Lady talking about her new prosthesis. The roleplay helps you not to focus too much on your voice and the "Ho Ho Ho" sound of Santa will push your voice back as described above.
Yawning is the perfect way you should breathe. The "opened up" feeling you get in your throat when yawning is the exact feeling you should get when singing. Try to add a melody to your yawning and apply this way of exhaling to your own singing. People can yawn very loudly because of the open state of their throat. Now try swallowing, which does the exact opposite of what yawning does. Compare the feeling of yawning (the exact thing you should do) and swallowing (the exact thing you shouldn't do) and apply it to your singing.
If all these don't help and you're still feeling tense, a vocal teacher is always a good way to improve your voice in any way.
EDIT: Doing further research, I've found this article which is very helpful to your question. It covers most of my points and gives more detailed information.