Vegas became the first player from Venezuela to win on tour after a three-man play-off with defending champion Bill Haas and fellow rookie Gary Woodland.
The trio had finished tied on 27 under after Haas closed with a 66 to catch the overnight joint-leaders, who both carded 69s.
Haas thought he had missed out after three-putting the par-five 18th for par, but after Woodland did make birdie Vegas also three-putted for bogey when par would have given him the title.
The trio returned to the 18th for the first play-off hole and Haas was eliminated after again only managing a par five, his opponents both holing from around five feet for birdie.
Vegas had almost snatched victory in dramatic fashion when his eagle chip from well right of the green caught the edge of the hole, but his chances looked to have disappeared on the second extra hole.
On the par-four 10th he pulled his tee shot into the water left of the fairway, but Woodland, having found the fairway, pushed his approach to the green into a bunker.
After taking a penalty drop, Vegas hit a superb recovery to 15ft and looked on as Woodland thinned his bunker shot across the green and then charged his par putt well past the hole.
That left Vegas with a putt to win and he made no mistake this time.
"This is a dream come true, just making it happen is something else," an emotional Vegas said.
Scotland's Martin Laird began the final round just two off the lead but slumped to a share of 22nd after a closing 74.
England's Brian Davis finished a shot ahead of Laird after a closing 67.