The best thing to do would be to pay off the debt and use the card to "live" on
This is correct. The idea is that you charge as much as you normally would on the new card, but only pay the minimum balance each month. The difference you put toward the student debt as an extra payment. If the credit card offers 0% APR then you are paying zero interest, so paying only the minimum is financially smart.
BUT ... the only catch is if you are caught in a situation where the 12 month teaser period runs out and you can't either pay off the card or roll it over to another one on good terms. Then all you have done is exchange student debt for credit card debt at a higher rate. This could happen if, for example you lose your job, get hurt and can't work, etc. That's fairly unlikely, just be aware that it's possible.
That said, 6% on only four thou isn't much, we're talking about less than two hundred considering you're paying it down over the course of the year. So it isn't a big deal either way.
But yes, you can do it and it can save you some.