You might actually get away with that. All the major components are in the same places.
However, the rear I/O is not going to match, so that might mean cutting out the back of the case. Front panel connectors might not be compatible, but you can bypass those if you have to and wire a power switch to the power supply's standby power and ground.
Chassis are cheap, pick up a $30 case new, or find an ATX case someone is selling locally. They often aren't worth the price to ship privately. Hit up some thrift stores and see what is around as well.