You might be able to tip the slab up, but it is difficult to brace it and get it to stay up. And you are right; it may crack.
Mudjacking is a better way, but it must be done by a pro with the proper equipment. Basically, it is pumping concrete mix under the slab and forcing it up in place. This applies a fairly uniform pressure that should not harm the slab, and when the pumped mud cures, it will stay in place.