Check road conditions and weather reports before you set out each morning. Even in Arizona and Utah you have a small chance of snowstorms or road closures due to whiteouts,accidents, or ice, so try to build in enough flexibility in your timing so that you could wait out a snowstorm for a day rather than feeling like you had to drive through it. There have been several previous posts about how to prepare for a winter road trip. It can be a beautiful time of year to travel, but it requires some planning and flexibility.
I would not recommend taking the Durango-Silverton-Ouray route in the winter unless the roads are absolutely dry. This is considered by many to be a "white knuckle" drive even in the summer. There are three major mountain passes and it is pretty likely that there will be icy spots at the very least, and road closures at the worst. There are extremely steep drop offs in places, and sections with very little shoulder and no guard rails, so there is no margin for error. Instead, travel from Cortez to Dolores and Telluride, where you should stop for a ride on the free gondola between Telluride and Mountain Village for awesome views, plus a walk around the historic village and maybe a visit to the museum. The scenery this way is, IMO, very nearly as dramatic as the Durango-Silverton-Ouray route. While still mountainous with a chance of snowy roads, the drive is not nearly as hazardous. If you'd like to see Ouray, just take the short detour south from Ridgway before heading back toward Montrose. When you get to Montrose, take a little detour east on Hwy 50 to the Black Canyon National Park. The rim drive is usually closed in the winter past Gunnison Point, but you'll be able to get some great views of the canyon. Also try to save time for Colorado National Monument at Fruita/Grand Junction. Another lovely place to visit is Redstone just south of Carbondale.
Keep two additional routes to Grand Junction in mind just in case. You could take the Unaweep Tabeguache Scenic Byway to Gateway and Grand Junction, or you could go through Utah on 191 to Moab and Arches National Park. I'd recommend the Telluride route as your first choice of route for the beautiful mountain scenery unless the weather happens to be really yucky.