Buying a car, and the dealer says it'll cost $1000 more if I pay cash

As the title suggest, dealer will charge me $1000 more if I don't finance through the dealer. Is there any reason why I couldn't finance the car, collect $1000 discount, and pay the car off tomorrow without paying any interest?

Edit: Wow. I didn’t expect so many responses! Thank you all for the advice. I’m negotiating with other dealers too, but getting the feeling that most dealers are going to charge more for cash payment… I hate buying cars, and your support is appreciated!