This even comes into play in games decided earlier, because if the home team is ahead by one run after eight and a half innings, they don’t bat at all in the bottom of the ninth. The home team’s margin of victory is “down-shifted” in all games because of this.
In any case, given the one-two punch of the college stats and the extra-inning stats, I’m satisfied beyond a reasonable doubt that batting last confers no advantage in baseball. This is counter-intuitive, because the team at bat can tailor its strategy according to how many runs it needs, and the team batting last has better information on that score. But the key point is that made by Quercus: The defense can also tailor its strategy (pitch selection, defensive positioning, which runner to attempt to put out), and the evidence suggests that defensive tailoring is almost exactly as important and effective as offensive tailoring.
By way of contrast, consider college football: In college football overtimes, going on offense last confers a clear advantage. In football also, both offense and defense tailor their strategy to the score, but the offense has more initiative–most importantly, they can kick a field goal or go for it on fourth down–and all the defense can do is react. In football, offensive tailoring is more important and effective. In baseball, not so.
As the old cliche goes, baseball is the only sport where the defense has the ball.