A Russian MiG-29 proved to be able to do that too over Georgia in May 2008. R-73 vs a Hermes-450 Israeli built UAV.
An Iraqi MiG-25 shot down an American Stinger armed MQ-1 in December 2002 (it fired back too unsuccesfully).
And you don't need any particular latest generation IIR missile or ARH missile or AESA radar...
Hezboallah Ababil UAV: F-16C/D/I with Python 4/5 (Advanced IR missiles)
Iranian Ababil UAV: F-16C and anything from 20mm to the latest AIM-120C
Pakistani Searcher II UAV: F-16A and AIM-9L
Georgian Hermes 450: MiG-29 and R-73
US MQ-1: MiG-25P/PD and R-40T or R-60
It is not a outstanding achievement actually. It's just a matter of finding them. As the fighter pilot sees them, they are toast.
...do not forget the MQ-9 downed by a F-15E over Afghanistan after the ground controllers lost contact with it. AIM-9M or X.
I'm not sure who we are going to be going up against that is going to be as proficient in the use of UAVs as the United States, but yes, the Viper can easily dispose of a Predator or Reaper. As the user above me stated, Manned aircraft can easily dispose of unmanned aircraft with AIM-9 Limas or Mikes or any sort of AAM for that matter. Drones don't really carry much in the way of countermeasures IIRC.