Good history of Bloombergs original primary victory
Apparently, it was seen as a longshot then too. Bloomberg was a lifetime Democrat and Democrats outnumbered Republicans 5:1 in NY. But when it became clear that he couldn't win the Democratic primary he secured the nomination of the Republican and Independence Part instead. The first through standard supplication to party bigshots but also strategic donations/supplications to neglected parts of the party apparatus (county-level and machine politic operators) and the second by promising support for non-partisan elections. The second party though insignificant won him 59k votes vs beating out the dems by only 35k votes.
He then picked off parts of the democratic electorate by running as a liberal and picking off hispinic voters (winning them 46 to 41%).
Of course he also spent $90 per vote, outspending say Ross Perot in total.