Dubrovnik has no shortage of stellar views, but if you are looking for one that's off the tourist path then go north of the Old City into the residential pedestrian walkways.
On Gornji Kono you'll find a paved area with park benches set up to enjoy a (likely private) sunset. See the attached map for the exact location.
A view is not the only thing you'll find here. Eploring the pedestrian walkways and seeing locals' backyards you'll get a glimpse of everyday life in Dubrovnik; notice the terraced yards, walled entrances, abundant gardens, and scooters parked at will.