Here's what Gurunet has, Father Steve:
fi·na·gle (fə-nâ'gəl)

v. Informal., -gled, -gling, -gles.

v.tr.
To obtain or achieve by indirect, usually deceitful methods: finagle a day off from work.
To cheat; swindle: shady stockbrokers who finagle their clients out of fortunes.
v.intr.
To use crafty, deceitful methods.

[Probably from dialectal fainaigue, to cheat.]

All that notwithstanding, when I think of someone finagling, I don't generally assign them motives that are actually sinister; more as someone who doesn't mind bending the rules a little, if it will get them what they want.

And, all of that notwithstanding, I would like to know what language that dialectal fainaigue comes from. And, is finagle related to the verb finesse?