Bates, Saaf and Colletta bring together Supergirl and Zatanna in Supergirl 7 (Oct 73), and wouldn't you know it, they wind up fighting over a guy.
Tony Martyn has gone missing in the Himilayas. Supergirl recalls how she saved him from going over a cliff in his car a year earlier, and was instantly smitten with him.
Similarly, Zatanna met him during a performance, and fell for the guy right away as well. He was seeing both of them before taking off on an expedition to prove that a local god was a fake.
Tony believed that some human was conning the villagers, pretending to be the demon Orgox. He vanished while looking for proof. Both Supergirl and Zatanna head on his trail, and they find a lost civilzation along the way.
But the women don't really care about that. When they learn that Tony was dating both of them, they become more concerned with trying to outdo the other. At least they don't wind up in a genuine catfight. The closest they get is when they are battling the demon, in the form of a yeti-type monster, and it vanishes, leaving them attacking each other. They decide they need to work together, and then Tony shows up, or so it seems.
It's really the demon making himself look like Tony, but Supergirl figures that out and the women take the demon down and rescue Tony. Neither winds up very happy, though, when they learn he has another girlfriend as well.
Zatanna had last appeared a couple of months earlier in Lois Lane, and is next seen a few months down the road in Green Arrow's series in Action Comics.