Yes it will work just fine.

Depending on what type of grass it is, you may still have problems. I built a raised bed in the spring, similar to your method: cut the grass real short & I then covered it with black plastic for about a month. Removed the plastic, laid down about 10 layers of newspaper, wet it down & added great soil. Then planted my veggies. About a month later, I had one or two blades of grass come up. When I pulled them, they had roots all the way to the bottom. Now, I have more & more grass - Bermuda Grass. All I can do is pull them up.

I went to our local newspaper and asked what they do with the ones that don't sell....she said "they're in the back in that huge recycling dumpster". I told her I needed them for landscaping and she allowed me to go dumpster diving! I filled up my van with bundled newspaper. And it was FREE!!!!

The public liabraries has retention time for newspapers and after that they send for recycling. You may be able to get large amounts from the library.

For raised bed I cut the grass very low, cover with 8 layers of newspaper sheets, wet it well, and cover with black plastic (I use the trash bags). After a week remove the plastic and put the soil on top of the paper. Some times I put garden fabric used to stop weeds. It works for St. Augustine grass in my zone. Bermuda is stubborn and have deep strong roots. But the principal should work because it kills the roots by starving them from light.

Jomoncon suggests putting the newspaper or unwaxed butcher paper after removing the plastic. The paper actually becomes compost with time.

Bermuda Grass will grow over 5', I know as I had a truck load of sand dumped over Bermuda Grass and it took a while but it grew all the way to the top of my pile. So good luck if you have Bermuda grass, The reason I know is when I finished using that load those roots were all the way to the original ground area where all the Bermuda grass was growing before.