Either way, you lose this one. Out is the call. This is one of the most enduring and commonly held beliefs that does not hold up when given scrutiny, and the rules of the game are examined. There is no such award of bases in the rules. There are have lots of base awards described in the rules.
Overthrown balls result in an award of two bases for every baserunner.
So, when the opposing team has a runner at first who is off with the crack of the bat, and your centrefielder makes a diving stab, catching the fly ball for the out, fires it toward first to get the runner who is well off the bag toward second, and throws the ball over everything into the bleachers, I don't want to see you when I send the runner to third base. One plus two is three after all. Otherwise he is out or can be put out. Other than home plate, first base is the only base that a runner can legally overrun without being in jeopardy of being put out by the defense. Runners at first base can turn any way they like after crossing the bag, just as long as no attempt is made to go to second.