"Saft" refers either to beverages made from boiled berries (which I've never tasted or seen, but my grandmother has described it) or the juice that's inside a fruit or meat or something. Once it's put in a glass and turned into a beverage, we call it "juice", same as in English. It's called "appelsinjuice" and "æblejuice", not "appelsinsaft" and "æblesaft".
The closest we get to referring to a beverage as "saft" (apart from that boiled berries-thing) is "saftevand", in which concentrated juices from fruits are mixed with water.
If you were visiting someone in Denmark and they asked what you'd like to drink and you asked "Har du saft?" (Which basically means... "Do you have liquids from biological organisms?" lol), they'd most likely look at you weird and say "Mener du saftevand?" ("Do you mean fruit concentrate that you have to mix with water?" Couldn't find an English translation for "saftevand"). You reply "No no, uhm... appelsinsaft?", they'd go "Nååårh, du mener juice!" ("Oooh, you mean juice!")
I'm just saying.