I weighed the same sort of thing when my wife and I bought our new van in June, whether to use savings or finance. We opted to finance and keep money in the bank for the very same reasons you're considering, just having extra money socked away for the unforeseen. The only thing that made this make financial sense is that our APR is low enough that this choice does not cost us a lot of money. The APR at my credit union on our savings is 3.3%, so that negates much of the financing cost.
With credit cards, it makes no sense financially unless you're able to transfer a balance to a low-intro or zero APR card. If you're stuck paying 18%+ interest, you're throwing a lot of money in the toilet. In that case it makes sense to pay off the balance ASAP, then if you have an emergency, surprise car repair, whatever, then just put that charge on the credit card which would have otherwise come out of your savings.
No sense accumulating interest on high rate credit cards while holding money in the bank on the chance you might need it. Pay the cards off, then use the cards again if you absolutely have to. Getting a low interest card or intro rate would allow you to pay off that card over a period of time.