It is probably something of an overstatement to say that Lincoln saved democracy. After all, both the North and the South believed that they were fighting for democracy. Each side saw democracy in a different light.
Regardless of whether Abraham Lincoln truly did save democracy, he clearly believed that he and the North were fighting to preserve it. We can see this as early as his first inaugural address. There, he constantly points to the fact that, as he sees it, the South has the ability to democratically change the Constitution if they wish to do so. Lincoln felt that the South was trying, as a minority of the population of the country as a whole, to impose its will on the whole country. It was for this reason that he said
The rule of a minority, as a permanent arrangement, is wholly inadmissible; so that, rejecting the majority principle, anarchy or despotism in some form is all that is left.
Lincoln was arguing that only a government by the majority of all of the people of a country could possibly be seen as democratic.
The Gettysburg Address is where we can see most clearly that Lincoln believed that he was fighting for democracy and against some other form of government. In this speech, Lincoln famously ended by arguing that the North was fighting the Civil War to ensure
that government of the people, by the people, for the people, shall not perish from the earth.
In these two excerpts from his speeches, Lincoln shows that he believes that he is fighting to preserve democracy.