So, here is my strategy for killing Lucky Yi, found just outside Halfhill.
This is a bit of a cheat I suppose, but it works fantastically every time using the pets in the order you see above, and with the abilities shown.
Firstly, use the Roach and its Apocalypse then swap it out for the Imp. All you now need to do is keep yourself alive til the Apocalypse hits and you’ll kill him.
My tactic is to cast Immolate on the Imp, to do some ticking damage then use Cauterize on cooldown to keep yourself alive. On the downtime between these two abilities, use Rush to do some damage. The Runt is there just in case of bad RNG or if you’re sleeping, but you shouldn’t need him.
After 15 rounds, the Apocalypse will hit and kill the not-so-Lucky Yi. Grats!