How did Abraham Lincoln hold the nation together after his election?

The key to understanding how Lincoln kept the nation together after his election is by examining two aspects of his presidency during that time. First, Lincoln's rhetoric during the war remained consistent in the fact that, above all else, the Union should remain intact. Northern abolitionists were calling for Southern blood, while Southern confederates demanded Lincoln gone. It is imperative to give appreciation to Lincoln's behavior, as he aimed to placate both sides. He acknowledged the evils of slavery but reassured the South by stating that his focus was to save the Union and not to necessarily destroy slavery:

"My paramount object in this struggle is to save the Union, and is not either to save or to destroy slavery. If I could save the Union without freeing any slave I would do it, and if I could save it by freeing all the slaves I would do it; and if I could save it by freeing some and leaving others alone I would also do that."—Abraham Lincoln to Horace Greeley, 1862

Another critical element to consider is the way he crafted the Emancipation Proclamation. Obviously from the quote above, in a letter written to Horace Greeley, abolitionists in the North were upset by the way Lincoln seemed to take a light approach toward slavery in the South. However, the Emancipation Proclamation eased that crowd's minds when it freed slaves in the rebellious states. Moreover, Lincoln was strategic in not freeing the slaves in the border states, out of the fear that they would join the confederacy. Once again, the rhetoric of Lincoln and the determination to preserve the Union allowed the nation to stay together in its darkest of times.
http://www.abrahamlincolnonline.org/lincoln/speeches/greeley.htm

Abraham Lincoln's election was the catalyst that launched the secession movement in the Deep South. Lincoln was not even on many Southern ballots and he won with a minority of the popular vote. Lincoln thought that the crisis in the South would blow over because it was only a minority of Southerners who demanded secession. Also, Lincoln did not want to alienate the border states. Lincoln promised to maintain all Federal installations including Fort Sumter. Lincoln did not ship weapons or reinforcements to Fort Sumter when South Carolina insisted on its surrender; rather, he shipped in food. He allowed the South to fire the first shot in the war, which it did when Pierre Beauregard's forces fired on the fort in April 1861.
At the war's start, Lincoln called for volunteers which led to the secession of the border states Virginia, North Carolina, and Tennessee. Maryland tried to secede but Lincoln instituted martial law there and even detained the pro-secessionist mayor of Baltimore in order to keep that state in the Union.