An arena has two entirely separate lines of sequences of mobs. As a hypothetical example, one sequence could start with two wolvers. Once you kill the two wolvers, then four lumbers spawn. When the four lumbers are dead, four kats spawn. When the four kats are dead, four zombies spawn.
The other sequence could start with two scuttlebots. When you kill the two scuttlebots, four retrodes spawn. When you kill the four retrodes, four gun puppies spawn. When you kill the four gun puppies, four devilites spawn.
There is also one last group that spawns when everything from both sequences is dead. Let's say that it's two trojans in this example.
The trick is to note that the two lines spawn successive mobs independently. When the arena starts, two wolvers and two scuttlebots will spawn. If you kill the last wolver and the last scuttlebot at the same time, then you'll immediately be hit with four lumbers and four retrodes all at once. That can get messy.
Suppose instead that you kill one scuttlebot, mentally mark the other, and don't touch the other scuttlebot. Instead, you kill the two wolvers. With the two wolvers dead, four lumbers spawn. Instead of four lumbers and four retrodes all at once, you just have four lumbers and one scuttlebot to deal with. As soon as you kill that scuttlebot, the four retrodes will spawn, but they won't show up until the scuttlebot is dead.
So you don't touch the scuttlebot and instead kill the four lumbers without harming the scuttlebot. That makes four kats spawn. Now you have four kats and one scuttlebot to deal with. But that's a lot easier than four kats and four retrodes, or four kats and four gun puppies.
Again, you leave the scuttlebot alone, and kill the four kats. That makes four zombies spawn. Nothing spawns immediately when you kill the four zombies, so you kill them and you're left with just the scuttlebot that spawned at the start of the arena.
Finally, you kill the scuttlebot. That makes the four retrodes spawn. But rather than four retrodes plus some lumbers, or four retrodes plus some kats, it's just four retrodes by themselves. You kill the four retrodes, and that makes the four gun puppies spawn. But again, it's four gun puppies by themselves, not four gun puppies plus something else. You kill the gun puppies and you get four devilites by themselves.
Finally, you kill the four devilites and because everything in both waves is gone, you get the two trojans. You kill the two trojans and end the arena without ever having to fight more than five mobs at a time, when it could easily have been eight.
It doesn't have to be one mob at the very start that you save. After killing the wolvers to make the four lumbers spawn, you could kill three lumbers and leave the fourth alone, while going through the other line. That way, you have to deal with four retrodes and a lumber, then four gun puppies and one lumber, and so forth. That's still easier than four gun puppies and four lumbers all at once.
The basic idea is, you'll constantly see a new wave of mobs spawn while the old one wasn't quite gone. Don't finish the old wave of mobs immediately, as that one is in a different sequence, and killing the last mob from it will make more of something else spawn. Rather, thin out the new wave before finishing off the old one. That way, whenever you get a new wave in one sequence of mobs, you don't have a bunch of mobs from the other sequence to deal with at the same time.
In particular, if you get down to just two mobs, don't kill them both at the same time. Kill one to make the next wave in one sequence spawn, but leave the other mob alone. That way, you have one mob plus one full wave, rather than two full waves to deal with at the same time.
Obviously, some mobs can't be left as a lone mob for long periods of time. If greavers show up, you have to kill them immediately, regardless of what it means about other mobs spawning. Gun puppies can be quite a nuisance to leave around, too. But you should think of the battle in terms of, that mob isn't hard to avoid, so I'll leave it alone and kill everything else first.
I just kill everything, works pretty well