Squeek, Vaul, and Captain Gallus (and others, of course!) have this dead on.
Sword Masters are the penultimate Close Combat unit. They love to be enveloped by weak troops (Low Toughness, Low Armor, Low Wounds). This allows them to have fantastic odds of generating lots of Active Combat Resolution. As you've seen, they can certainly overpower Static Combat Resolution in the right circumstances.
As Vaul said, all you did was feed "meat to the grinder." As Vaul and Squeek disclosed, your best bet would be to just tie them up with a cheap unit to the front, or sick some assassins and their cohorts on the Sword Masters to the flank (made worthwhile from the extra combat resolution).
Your overall goal is to deny them the opportunity to optimize their potential. The High Elf players want their Sword Masters stuck in combat with a boat-load of junk units. Magic them, Shoot them, or tie them up to deny them this. There's also plenty more ways to confront the issue at hand, and as a Skaven player I'd imagine you could have plenty up your sleeve to surprise most stalwart Elf players!
White Lions are just as tough to bring down. They're Stubborn with the same save in CC, and hit like a ton of bricks. They specialize in fighting High Armor or Toughness troops. Deny them that ability, and you're golden. Oh, and Static Combat Resolution (they'll get +1 or +2 for ranks, +1 for banner, probably won't outnumber or anything of the like, and can deal out six attacks to the front maximum). Unleash the Hordes!
Phoenix Guard are the hardest to kill of all. Their Ward Save and Heavy Armor tends to keep them around for a long time. They cannot generate offensive damage, though. Their Halberds just aren't enough to tackle Hordes. You'll have to beat these guys with Static Combat Resolution here. Just don't get outnumbered.
I know you said you beat the High Elves before. What did you do to win? Since you won, why do you see it necessary to complain about the Sword Masters? Sure, they're nasty, but you overcame the rest of the army after all was said and done. We can all agree that they were a points sink in your specific situation, so did that have something to do with it, NagashKhemmler? Or did the rest of your tactics allow you to smash the Asur? Fill us in; we need some context so we can help you out better!
I hope this and everyone else's comments help, NagashKhemmler. Happy Hunting!