I have successfully "controlled" (not necessarily eliminated) a mulch bed that was overgrown with Bermuda grass. I'm in zone 7, with lots of clay. My first attempt was using commercial fabric, but as the other posts explain, that causes an extreme head- (and back-) ache in subsequent years, when all of the fabric needs to be removed because the grass has totally penetrated it everywhere.

What actually worked, was changing the clay soil into loamy soil by adding significant amounts of Perlite and Peat Moss, and then covering with mulch. Now, when the Bermuda Grass dares to infiltrate the mulch bed, I simply lift it out with a Garden Fork. It's that easy. Occasionally, a runner will lie deeper than the loosened soil and reemerge-- but it's easy to loosen the soil in that location (by adding more Perlite and Peat Moss) a bit deeper until most or all of the runner is removed.

The only hard part was in the vicinity of bushes that had been infiltrated with runners. in some cases, it was actually easier to remove the bushes (if small enough), clean out the runners, and then replant. In other cases, it was a matter of tug-of-war with the runners. While not all were removed, they can be controlled by occasional weeding; and if any spread to the open areas, it is easy to remove with the Fork. Over time, the mulch degrades and further improves the soil.