How to get a Most Wanted car after you have raced it and won it in Need For Speed Most Wanted (2012)? LINK
After the race ends the opponent's vehicle drives away. Do I have to follow it or is there certain actions I need to perform to claim the vehicle?
You have to shut it down. Right after finishing a most wanted race, you have to crash/total it. Even if you can't do it it's fine. You will see the most wanted car roaming around the map, it will be highlited by black icon on map.Just crash it right then and you'll get it.
After winning against a "Most Wanted" driver in Need for Speed: Most Wanted (2012) they will drive away and you must "take down" their vehicle to receive it as an award. Take-downs work the same way as you would destroy a police vehicle or opposing racer in the game - slam into their vehicle hard enough to destroy it.Link
If you aren't able to take down the Most Wanted driver after the race they may manage to escape and disappear from your minimap. If this happens, continue exploring/free roaming the in the open world. The Most Wanted racer will randomly spawn in the open world and you will have another opportunity to take them down. From my experience, they spawn about as frequently as Fairhaven's police cars.
If you are again unable to destroy their vehicle they will again despawn and you will have to wait for them to spawn again. Once the vehicle is destroyed you can find the Most Wanted car in parked in open world, like other drivable vehicles are, as well as being able to drive it using the EasyDrive menu.
You can read a little more about the Most Wanted list from this wikia article
