The only answer to quack gras, as you have discovered, is Round up. If opposed to Round Up, your only other option is to dig it out weekly. But it will come back.

Glyphosate, contrary to popular opinion, will not control quack grass although it may slow it some for a year. But every one I know that sprays quack grass with any kind of poison does so every year. The only way to keep the rhizomes from your garden is with a barrier those rhizomes cannot get passed. The plastic carpet runner is a clear plastic you would put down over your carpet for people to walk on so your carpet does not get dirty. I have seen it available in 18 inch wide rolls that you could cut in half, but even then it is more expensive than the aluminum flashing. While the initial expense of either of these items is high I find the long term expense is considerably less than purchasing plant poison every year.

Glyphosate can control quackgrass completely, however it will not usually control it with one application and mid range rates must be used. Lower rates will provide only suppression. Follow up, not annual applications are the key. The nature of quackgrass physiology allows it to resist single applications.

This site explains the issue well:
"One problem with using glyphosate on quackgrass is that up to 95% of the lateral buds on the rhizomes are dormant even though the plant is actively growing. Since herbicides are translocated from the leaves to actively growing plant tissue, after about 7 days the glyphosate degrades and the dormant lateral buds will start to grow new shoots. It may take more than one application to completely eradicate quackgrass."

Here is a link that might be useful: Quackgrass management

I have a problem with quackgrass in a newly seeded lawn. Apparently, the straw mesh used to protect the seed from washing away contained quackgrass seeds. My landscaper applied a herbicide that will kill crabgrass to no effect. He then applied Beacon (primisulferon) which seems to be working. The quackgrass is slowly turning brown without affecting the lawn grasses. I'm assured this herbicide will destroy the quackgrass. Beacon is not available to the general public. I think you have to be licensed to use it.

Research at Iowa State University a few years ago and lately and many other universities has shown that the glyphosates do not do a very good job of contol on Quack Grass and often needs to be applied 4 times a year to keep it under a bit of control. Where the glyphosates have been used to control Quack Grass around here I have seen the stuff growing back about 3 to 6 weeks after application.
Tilling any soil that contains Quack Grass roots will simply cut them into multiple pieces that will then root and grow more grass plants, tilling a soil containing Quack Grass roots simply spreads the stuff over a wider area.