If the cushion is actually soaked, rather than just a wet surface, I'd try the following:
1. Remove cover and set aside.
2. If the cushion case is zippered as well, then remove the case and put it with the cover. If it's sewn shut, I wouldn't bother with ripping a seam unless it's foam filler that I'll need to discard and replace.
3. Look over the filler: a) If it's foam, you pretty much can't remove urine from it, fortunately it's cheap. Take the dimensions and go to an upholsterer or fabric shop, they can cut you a fresh slab of foam. b) If it's fiber fill, polyester or cotton, you can wash it and I'll cover that in a bit. c) If it's good quality cotton batting in layers, open a seam, untie the layers and you'll be washing them with the cushion and cover. Take the measurements of any foam layer to discard and replace if you encounter any. d) If it's the really good stuff such as wool and/or horsehair, you'll need to look up the proper cleaning method, because it certainly won't be what I'm recommending for the rest.
4. Take a large bucket, deep sink or tub which will fit your cover, case, and fiber fill cushion or batting if you have any, with room to circulate. Fill with lukewarm water and agitate gently.
5. Discard water, refill with a detergent of your choice (by detergent, I mean a straight detergent in the chemical sense, not any old laundry detergent, if you're in doubt, try grated plain soap). I'd add Oxiclean, borax, and washing soda, especially the borax which works wonders on urine odors and will remove yellow coloration, but spot check with each first, and don't use borax on wool or silk. Agitate thoroughly by hand (or with a clean plunger if you have one).
6. Rinse thoroughly.
7. Hang dry fabric and batting layers out of direct sunlight, but where there's good circulation. Put fiber fill cushions and throw them on no heat, tumble dry with dryer or tennis balls to beat the fluff back into it.
8. Reassemble when dry.
It sounds onerous, but it's really not that time consuming, except for the time spent drying. I give the treatment to a few items in the household once a year "whether they need it or no." In your case, I'd just be happy he didn't get the non-removable upholstery, to me that's in the realm of professionals only, I won't touch it.