Although it is fancy to think of antivenom as a poison for the poison, technically it is a poison only for the poison!
Antivenom is developed in horses (equine) by hyperimmunizing them against the particular venom at a non-lethal dose, followed by collecting the serum. Venoms are complex and have many enzymes and hence are proteins. So, antivenoms are just antibodies against the venom proteins.
Moreover, the antivenoms are classified into monovalent and polyvalent. This signifies whether the sera was prepared by hyperimmunizing the animal with the venom of a single species or many common venomous snake species together.
Usually, a snake bite is an emergency and the victim or the attendants rarely identify the snakes (In our place, patients sometimes to our surprise, even bring the killed snake to the casualty!). Hence, to avoid confusion polyvalent sera are used.
I've written all this just to make an impression that polyvalent sera also carry antibodies against the wrong snake too, but at the end of the day, it's the right antibody that does the job.
As mentioned in the comments, apart from occasional anaphylactic reactions or serum sickness, they are pretty safe (cost/benefit) to use.