Nope, you absolutely need to use an 11 speed chain and cassette. Pretty much the only reusable drivetrain part is the crankset. You can install a new free hub body onto your dt hub. DT hubs come in star ratchet style internals of 3 pawal internals--you can get either upgraded to 11 speed, but you have to get it right, the 3 pawl freehub bodies are like 60 bucks and the star ratchet are like 80.

If you have a bike with 10 speed Red that has been ridden regularly since it's purchase, I'd maybe consider replacing both the derailleurs outright--the pivot points on older derailleurs do wear and while they adjust pretty well it effects how well they ultimately shift on the road. Replacing the FD with a 10sp Shimano FD will also probably shift better as the 10sp Red FDs were built excessively light. With new/quality cables and housing and good quality 10sp Red typically shifts very well.