I don't know of any off the top of my head who won their very first tournament, but I also don't have a memory bank for the initial results of many players..
For what it's worth, I know that Jose Acasuso of Argentina reached the finals of his very first ATP tournament. He fell short of winning it though. Buenos Aires 2001, if anybody is interested:
Round Opponent W/L Score
R32 Hicham Arazi (MAR) W 2-6 7-6(3) 6-4
R16 Albert Portas (ESP) W 6-4 7-5
QF Franco Squillari (ARG) W 6-7(3) 6-4 6-4
SF Gaston Gaudio (ARG) W 3-6 6-1 6-3
F Gustavo Kuerten (BRA) L 1-6 3-6
Not bad for your first ATP tournament, huh? Keep in mind that Arazi was posting some decent results back then, and not lingering out of the top 100.
That doesn't answer your question, but it makes for a good sub topic. Another good subtopic, is how many tournaments did it take the top players to win their first title?
I know of a few off the top of my head:
I believe that Roddick took 11 tournaments.
Ferrero took 5.