Start a race in Quick Race mode and choose the car you want to unlock. Start the race, but quit before finishing it and save your game. Now load your save file and head to Career mode. You'll find the car available, though you still have to purchase it from there.
Unlock the following cars by defeating the corresponding people and finishing in first place:
CAR - PERSON Aston Martin DB9 - EX Audi TT 3.2 Quattro - Poorboy Ford GT - Clutch Ford Mustang GT - Poorboy Mazda RX-7 - Arjen Mazda RX-8 - Perimeter Sprint 3 in Lucky Sevens Mercedes-Benz SL65 AMG - Clutch Mitsubishi Eclipse GT - Perimeter Sprint 3 in Lucky Sevens Mitsubishi Lancer Evolution IX - Layla Porsche 911 Carrera S - MK Subaru Impreza WRX STI - Layla Toyota MR2 - Arjen Toyota Supra - MK