To easily beat DARIUS in his CANYON DUEL, get a TUNER car. Hold out until you are around 60 - 65%. After the 2 HAIRPINS, you will head onto a straight that gets your "follow" meter into the ORANGE zone (try to stay in the HIGH-ORANGE area or better.) Then you'll encounter a "WIDE" TURN WITHOUT a BREAKABLE WALL. DARIUS will hit the BRAKE at this corner and take the INSIDE. You, however, should stay on the OUTSIDE (I know it doesn't make much sense, but it works!) If done right, you should PASS DARIUS! The moment you pass him, get in FRONT of him and BLOCK him from passing. Hold out for 10 more seconds and you win!

P.S. Sometimes, it might not work on the first try, so
don't get frustrated. (It took me about 10 times to
work.)