Casey charged from five strokes behind in the final round to overhaul Woods and Patrick Reed by one shot in a compelling finale at the Innisbrook Resort in Palm Harbor.
"It's been too long," the 40-year-old Englishman said after ending a nine-year PGA Tour victory drought following his maiden success at the 2009 Houston Open.
"Very satisfying ... Probably more satisfying the fact that it was on a week where Tiger played some good golf and got to see some amazing stuff and hear the roars and it just was a great week, the buzz, everything about it.
"I loved it. I played great. I couldn't be happier."
The Englishman also enhanced his chances of making the European team for this autumn's Ryder Cup against the U.S. in France.
Casey last year rejoined the European Tour so he would be eligible. He has won 13 times on that circuit.
Once ranked as high as third in the world, Casey has long been regarded a premier ball-striker. He says improved putting had solved the final piece of the puzzle.
No wonder he lapped it up when Woods congratulated him on his victory.
"It's the only time he's congratulated me immediately after a victory," Casey said after shooting 65 to finish at 10-under 274.
"Normally it's the other way around. That's something special. I actually thought he was going to win today before the round started. I thought it was just teed up beautifully for him."
Casey played this week with a heavy heart following the recent death of the wife of a member of the Burhill Golf Club in Surrey, where he spent many of his formative years.
With the April 5-8 Masters looming, Casey has pulled out of this week's Bay Hill Invitational and instead will return to England for the funeral.
Then he will be back in action at the World Golf Championships-Match Play event in Austin, Texas, before taking two weeks off to prepare for the first major of the year at Augusta National.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Ian Ransom)