This is a toughie to answer because I think it depends very much on the wider nature of the person involved. In my own case I was a fairly emotionally sensitive child but I was raised in a "crying is for weaklings" environment. I learned to control it largely through the social embarrassment which follows public displays of tears in such an environment.
Let me say right now that unlike many people from such backgrounds I do not think this is a particularly bad thing. To this day I find public weeping to be an embarrassment and something rather shameful in an adult. Not in myself, because I don't do it: in others. I expect you will receive some answers saying, basically, "don't fight it, it's natural, let the tears flow". That's an opinion, certainly, but not one I share and from the tone of your question I suspect its not one you share either.
So, I think that those who are prone to tears and who wish to control them need to find the counter-emotion that works best. In my case it was social embarrassment and self-disgust (borne of peer pressure to some degree, but not entirely). I know of other people who have successfully used anger to fight back tears, but there are obvious side-risks with that one. Yet other people of my acquaintance tell me that they can make their minds go blank and use an almost meditation-like technique to simply empty their minds of thoughts until the emotional tide recedes. Try closing your eyes and taking deep, steady breaths while trying to clear your mind in this way.
So, some sort of displacement thinking is what seems to work. Find something you can think about or make yourself feel which causes other emotions to shore up the dam.
I can't remember where, but I've read that abstract thought is a good antidote to tears. I wish I could find the reference so as to explain the phenomenon. Try memorizing and reciting an abstract poem in your mind, or using your imagination to construct an abstract visual scene to focus on.