mordecai with his shield passing skill

EDIT !:
That's definitely the most clear-cut way to beat them... if you're Mordecai. For the other three, it's best to use a shock weapon that has both a high DPS and a good proc chance. I would normally suggest a scoped revolver... but shock revolvers are usually terrible. I was lucky enough to find an S&S Orion before reaching the Eridian Promontory, so I used that.

Take note that guardians have heads, just like other humanoid enemies; shooting their heads WILL yield critical hits, but of course, it's much more difficult to do that with a guardian. I got my orion early in playthrough 1... I think it's somewhere around level 20, and I would say it would take that gun 4-6 criticals to kill an Arch Guardian. While that might sound underwhelming, the Orion has a clip size of 15 (or maybe it varies, I don't remember, but it's very large either way) and its fire rate and reasonable recoil mean that you can pop off one badass wave of shock before reloading.

If you're not keen on sniper rifles for dealing with them, you could try a shock SMG, any shock combat rifle, or a shock shotgun, and get up close. A spray and pray offense while aiming in the general direction of the head will yield at least a few critical hits, simply because of the large number of projectiles being fired. Always make sure you have an escape path, but if the thing is close to dead, let it cripple you so you can gain a second wind, rather than risking a cripple when you're too far away to hit it.

EDIT 2:
An interesting comment from http://borderlands.wikia.com/wiki/Vengeance concering the Vengeance line of repeater pistols:and its special effect to bypass shields:

The special effect is currently bugged:

If the Vengeance is not intially equipped when the game starts, its special ability will not take effect.

If the Vengeance is equipped to one of the four weapon slots when the game starts, all four equipped weapons will always have a chance to bypass shields.

If the Vengeance is replaced with a different weapon after the initial load, all weapons will still have a chance to bypass shields.

Can also carry over to driver's front turret, plus it might pass to Rocket and Turret passenger seats. Not yet tested for passenger weapon. (Person with the gun only.)[citation needed]

Drmaxman 19:39, April 25, 2012 (UTC)