[okie01]

How can someone who was not eligible to be in the race finish first?

The Obama-Biden ticket was elected. So far as we know, Biden is eligible -- there being no IQ test for president.

[seton89]

Biden was elected VP.

Who was elected president? The eligible candidate with a majority of valid electoral votes - McCain.

Voting in the EC is not by ticket. The P and VP are voted on separately.