You could use "I invited them for my birthday party".
You are going away for the weekend with some friends for your birthday. On the Saturday evening, you plan to have a big birthday party. You have invited an extra twenty people to come just on the Saturday evening. One of your friends doesn't really understand what's going on and the following exchange takes place:
Friend - Someone told me another twenty people are coming on Saturday evening.
You - Yes, that's right.
Friend - I don't understand. Why did you invite them?
You - I invited them for my birthday party.
It's true that what you have done is invite those twenty people to your birthday party but you have invited them to the venue for (meaning "in order to attend") your birthday party.