During the Second World War, the British De Havilland Mosquito light bomber achieved success in raids through its sheer speed. Traveling at up to 392 mph, it could outrun virtually all piston-engined interceptors, removing the need for it to carry defensive weaponry. Later in the war though, German jets like the ME-262 could travel at speeds in excess of 500 mph, seemingly making the Mosquito obsolete. How did the Mosquito compete with these jets successfully, despite the obsolescence of its only major advantage?
The Mosquito was a great aircraft but it was not faster than Germany's Messerschmitt ME-109 and Focke-Wulf Fw 190 piston powered fighter aircraft.
It also did not successfully compete with the jet powered Messerschmitt Me 262. The limited numbers of ME-262 were probably the only reason the Mosquito could survive in the same airspace.
On 26 July 1944 the first aircraft to be shot down by a ME-262 was in fact a Mosquito: Mosquito Operational History
"a 262 A-1a, caught and severely damaged the Mosquito in a pursuit. The Mosquito, which managed to escape into cloud, later crashed at its destination airfield in Italy, and the airframe was written off"
There's a pretty good account of two of the early encounters between photo recon Mosquitos and an ME-262 at this link: http://www.mossie.org/stories/Norman_Malayney_2.htm
In both cases while the ME-262s were about 50 mph faster than the Mosquitos, because the Mosquito was much more maneuverable than the ME-262 both survived.
By the time that the ME-262 became operational, there was a huge lack of fuel, oil and resources in germany. They also lacked trained, experienced pilots. This means that missions in the ME-262 were limited, and they did not usually go up to shoot down photo-recon mosquitos. Instead the ME-262 and other german fighters of the time were reserved for anti-bombing missions were they tried to shoot down enemy bombers. This means that encounters between the Mosquito and the ME-262 were not that common, and if they found each other, the ME-262 would usually win.
the first encounter between the Mosquito and the Me262 was reported by the Mosquito crew (Wall and Lobban) on 25th of July 1944, while the Me262 pilot Scheiber reported the encounter on 26th July 1944. Scheiber reported the Mosquito as a kill but the the latter survived and landed safely at a friendly base in Italy.
available records do not specify if said Mosquito was able to operate again, but damage was described as non-lethal. the navigator did mention that the "outer dome" was blown out, but if it could fly from Germany to Italy then likely the damage was not enough to warrant a kill. it probably flew again.