How/why do they not split?
Perhaps Douglas makes Breckinridge his VP and makes a commitment to be "reasonable" on the question of slavery in the territories, thus giving ammunition for rational Southern Democrats to use against the firebrands (who probably split the party because they genuinely wanted an excuse to secede).
If we use the (admittedly unconvincing) assumption that all Breckinridge votes IOTL would go to Douglas ITTL, the Democrats would win three states that Lincoln won IOTL: California (4 EVs), Oregon (3 EVs), New Jersey (7 EVs). They would also win the three states that the Constitutional Union ticket won IOTL: Kentucky and Tennessee (12 EVs each) and Virginia (15 EVs).
All this, combined with the 72 electoral votes Breckinridge won IOTL, gives the Democratic ticket 125 electoral votes to Lincoln's 166. So Lincoln would still win the election.
Of course, this doesn't take into account the huge number of butterflies that would be released by a united Democratic ticket. It's possible that the Democrats might have emerged victorious. But it certainly would have been a lot closer.
EDIT: Forgot Missouri, which Douglas won IOTL. A unified Democratic ticket would therefore have gotten 134 EVs, which would still not be enough to beat Lincoln.