How did the De Havilland Mosquito compete with contemporary jets?
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"