The Teen Titans fight an alien in issue 12 (Nov/Dec 67), thanks to Haney, Novick and Cardy.
We get to see a little bit more of Titans Lair at the top of the story, chiefly that it has a pool, for Aqualad. The story centres on Wonder Girl's favourite radio dj, who is broadcasting (somehow) from a ship orbiting the Earth. This tale gets so preposterous it's best not to even think about the reality of any of it. Wonder Girl picks up that the dj is repeating the words May and Day, and figures out that he needs help.
But before they can do anything about that, Robin, Kid Flash, Wonder Girl and Aqualad wind up dealing with the Deliverer, an Earth criminal who steals things and delivers them to his clients. Good name for that. The Deliverer has a sort of laser ray gun, and uses it to cut the carved part out of Mt. Rushmore, and send it into the air. The Teen Titans manage to stop that, and a later attempt to send the Sphinx into the sky.
Then they just decide to get onto a rocket and head out to intercept the orbit of the dj, who sure enough is being menaced by an alien, the client of the Deliverer, who wants to loot Earth's big monuments. The cover image captures the climax of the story, as the Titans defeat the alien and save the dj.