What is it called when you use someone's exact words against them?

There are a number of expressions that suggest using a person's own words, methods, or weapons against them.

• Giving them a taste of their own medicine

This means doing to someone what they have either done to you or are known to do.

• Poetic justice

This means that a person's end or punishment was fitting for their actions.

• Turning the tables

Not so much using someone's weapon against them, but reversing fortunes, perhaps turning a disadvantage someone gave you into an advantage against them.

• Comeuppance

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:

• Give someone enough rope to hang himself

A slightly different angle, but this means to allow someone to bring about their own downfall.

• They backed themselves into a corner

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.

On tvtropes.org the example you give falls under the trope "ironic echo"

https://tvtropes.org/pmwiki/pmwiki.php/Main/IronicEcho

I've heard: You opened the door, I just walked in.