Don't worry! My prom dress was strapless with a ribbon laced up the back so my bra would have shown through too. I got a strapless, backless bra! They work by having a sticky film on the inside of the cups so you just place them in position and they stay on all night. I don't know what country you're from, but I know you can find them in Primark in the UK. I got mine from eBay though; they're even reusable, you just wash the inside of the cups and apply a spray that usually comes with them and they become sticky again. I really would recommend them!
I'm not sure you can see, but this is the best picture I found. My dress was only a little open, but there was no trace of bra:
