How was Lincoln able to win the 1860 election despite not being on southern ballots?

He won the election because he took advantage of the split in the Democratic Party and won a plurality of the vote.

Explanation:

The passage of the Civil Rights Act of 1964 was a significant event in converting the Deep South to the Republican Party; in that year most Senatorial Republicans supported the Act (most of the opposition came from Southern Democrats), but the Republican Party nominated for the Presidency Arizona Senator Barry.

(I couldn't really answer your question, but I wrote about what I know.)