I recently went to a Nissan dealership to buy a used car. They are claiming that they can offer me a price that is over $2000 less if I finance with them, instead of paying in cash. Is this a legitimate practice? I understand that they get dealer kickbacks, but they also don't get their money upfront.
It is a legitimate practice. The dealers do get the loan money "up front" because they're not holding the loan themselves; they promptly sell it to someone else or (more commonly) just act as salesmen for a lending institution and take their profit as commission or origination fees.
The combined deal is often not a good choice for the consumer, though. Remember that the dealer's goal is to close a sale with maximum profit. If they're offering to drop the price $2k, they either didn't expect to actually get that price in the first place, or expect at least $2k of profit from the loan, or some combination of these.
Standard advice is to negotiate price, loan, and trade-in separately. First get the dealer's best price on the car, compare it to other dealer and other cars, and walk away if you don't like their offer. Repeat for the loan, checking the dealer's offer against banks/credit unions available to you. If you have an older car to unload, get quotes for it and consider whether you might do better selling it yourself.
Standard unsolicited plug for Consumef Reports' "car facts" service, if you're buying a new car (which isn't usually the best option; late-model used is generally a better value). For a small fee, they can tell you what the dealer's real cost of a car is, after all the hidden incentives and rebates. That lets you negotiate directly on how much profit they need on this sale... and focuses their attention on the fact that the time they spend haggling with you is time they could be using to sell the next one.
Simply walking into the dealer with this printout in your hand cuts out a lot of nonsense. The one time I bought new, I basically walked in and said "It's the end of the model year. I'll give you $500 profit to take one of those off your hands before the new ones come in, if you've got one configured the way I want it." Closed the deal on the spot; the only concession I had to make was on color.
It doesn't always work; some salesmen are idiots. In that case you walk away and try another dealer.
(I am not affiliated in any way with CU or the automotive or lending industries, except as customer. And, yes, this touch keyboard is typo-prone.)
With new cars it's usually the other way around: finance at a low APR or get cash back when you buy it outright. With used cars you usually don't know how much they have invested in the car, so it's more difficult to know how low they're willing to go. Regardless, I do think it's odd that they would knock 2K off the price if you finance with them, but not if you pay cash. The only reason they would do that is if they intend to make at least 2K in interest over the life of the loan, but they have no way of guaranteeing you won't refi. Therefore, I suspect they are bluffing and would probably close the deal if you wrote them a check (or put the cash on the table) for 2K less.
However, if they won't budge and will only knock off 2K if you finance, you could finance and pay it off in full a week later. Just make sure they don't have any hidden origination fees or pay-off-early fees.