Till it no matter what you do. I would either scrape the existing stuff off, or bring in a sod cutter to get it out.
then, till it a few inches. Add compost Till that in, then roll it. Rake to take off high spots/fill in low spots. Make sure to roll it, or you will have settling problems...
Then sod, and roll again. You want to press that dirt layer on the back of the sod down into your freshly prepared soil.
Tilling is good; it breaks up the soil so the new roots can get established. The compost helps get nutrients to those new roots.
It is also helpful to till, so water can get in the soil. If you don't, you'll have a soil horizon. Water will want to follow that horizon(travel horizontal, instead of soaking down and in).
Remember too, to water often at first while roots establish. Then water less often, but water deep. Check with you local Ag extension or Master Gardener office. They can get you the transevaporation rates. This will tell you how much water you need each week. From there, you can tailer your irrigation schedule. Remember, watering less often, but getting a deep watering promotes a deeper root bed, which is healthier and less prone to disease and shock from heat ect.