There are a number of expressions that suggest using a person's own words, methods, or weapons against them.
This means doing to someone what they have either done to you or are known to do.
This means that a person's end or punishment was fitting for their actions.
Not so much using someone's weapon against them, but reversing fortunes, perhaps turning a disadvantage someone gave you into an advantage against them.
This simply means that a person received "what was coming to them"; that unpleasant words or behaviour resulted in the most likely or expected unpleasant outcome.
Perhaps less so:
A slightly different angle, but this means to allow someone to bring about their own downfall.
This means that the person by their own careless words got themselves into difficulty or lost an argument.
Depending on how you are using this, I thought it might also be worth suggesting that such a situation, where somebody is beaten by their own words, may be an example of irony. To be a true example of irony, the result must be the reverse of what was to be expected. So, if a person said something expecting his statement to win an argument, but his words actually caused his downfall, this would fit.