Clear the Catacombs

I used this strategy:

Iron Starlette (1/1/1)
Nexus Whelpling (1/2/2)
Clockwork Gnome (1/1/1)

Against Ancient Catacomb Spider

1. Wind-Up (first use)
2. Powerball (if it misses, it doesn't matter)
3. Supercharge
4. Wind-Up (second use)

Ancient Catacomb Spider should die at this point. If the 2nd Wind-Up misses, abandon the fight and start again. The order of this rotation is important, otherwise your Wind-Up will miss due to spider being burrowed.

Against Catacomb Bat

You can try to get a few hits in with the Iron Starlette. Sometimes you're lucky and Iron Starlette can even kill it. Once Starlette dies, switch to Nexus Whelpling, cast Arcane Storm, then Mana Surge. Fill in with Tailsweep.

Against Catacomb Snake

Switch to Clockwork Gnome. Cast Build Turret on CD, fill in with Metal Fist. If your Gnome drops to about 50% HP use Repair. Don't worry if the heal kicks in after the Gnome's passive ress, this way your Gnome should be back up to about 900 HP and can easily finish off the snake.

I used:
Darkmoon zeppelin
Teroclaw Hatchling
Any level pet 1+

Turn 1: Bombing run
Turn 2 Missile
Turn 3: Decoy
Turn 4+ Missile until the Ancient Catabomb Spider dies
Catacomb Bat comes in

Turn 1: swap to your Teroclaw Hatchling
Turn 2: Dodge
Turn 3: Alpha Strike
Turn 4: Nature's Ward
Turn 5+ Alpha Strike until the Catacomb Bat dies
Catacomb Snake comes in

Turn 1: Swap to your Level Pet
Turn 2: Swap to your Darkmoon Zeppelin
Turn 3: Decoy
Turn 4: Bombing Run
Turn 5:Missile until the fight is done

Worst case bring back your Teroclaw Hatchling and use Dodge + Alpha Strike.

How to get there:

The "Ancient Catacomb Egg", which starts the pet battle, is in a small cave West of the Crimson Thicket flight master. The cave entrance is at the end of a downward staircase (cave opening 59.4, 43.1). To get to the staircase, travel southwest down the main road from the flight master and turn left just before the third lamppost. The stairs entrance cannot be seen from the road (staircase starts at 59.1, 44.8).