Strip the grass,perhaps with a rented sod cutter.
Stack the pieces upside down, cover with a heavy tarp, and let the stack compost for at least a year then add back to the garden.

If you only till it under, you'll have LOTS more grass.

Thanks for the great ideas.

Mac

If you smother the grass, it takes a few months to make sure it is dead, but then does not need to be removed. Not ready to use as quickly, but the easiest to do, and no loss of the top soil and organic material of the dead grass. As a compromise, you could sit potted plants in the area while waiting to make sure the grass is dead.