Thankfully, "Fast and Furious" expert Craig Lieberman has an explanation. Lieberman was the chief technical director for the first two movies and has loads of inside knowledge about the cars used for filming. For this question, there are three schools of thought.
The first is to take each hero car on paper and see how they match up. The engine supposedly in the 1970 Charger was a 526-cubic inch supercharged V-8 running alcohol to make around 900 horsepower. The Supra featured a standard 2JZ inline-6 with a T-66 turbocharger. Lieberman estimates the weights of both cars to be 3,900 pounds for the Supra (remember, it had a lot of "tuner" stuff not meant for drag racing) and 3,600 pounds for the Charger. Also recall the Supra had a wet shot of nitrous that, best case scenario, helped it make 800 hp.
Drag race scene from 'The Fast and the Furious'
Based on this scenario, the Charger is the winner. There's simply no way the Supra would have crossed the line first without proper drag slicks, which the Charger featured.
Now, scenario number two is what the cars onscreen were actually running. Since the hero cars weren't used in this scene, they weren't making all the horsepower from scenario one. It's conceivable the Supra would be the winner in this case. Lieberman said most of the Chargers used likely had the 318-cubic inch V-8 and none of them were running superchargers.
However, the final scenario is the one the film's producers wanted all along: neither of the cars wins. The producers knew Brian and the Supra were the heroes, while Dom and the Charger were the anti-heroes. Neither of them were supposed to win this race. Lieberman wraps things up and says if you slow the scene down frame by frame, it looks as if the Supra touches down first, but maybe by the slimmest of margins. But, Dominic Toretto famously said, "winning is winning."