Casey carded a final round of 65 on the Copperhead Course at Innisbrook Resort to set a clubhouse target of 10 under par which American Ryder Cup star Patrick Reed looked set to match before he bogeyed the final hole.
Woods was two behind with three holes to play but holed from 40 feet for an unlikely birdie on the 17th and had a similar distance on the last to force a play-off.
But the 14-time major winner, playing just his fourth PGA Tour event since undergoing spinal fusion surgery last April, was unable to repeat the feat and had to settle for a share of second place.
Matt Wallace believes he played some of the best golf of his life to secure his second European Tour title at the Hero Indian Open. Wallace defeated Andrew Johnston in a play-off at the DLF Golf and Country Club in New Delhi after the pair had posted final rounds of 68 and 66 respectively to finish on 11 under par.
The 27-year-old won it at the first extra hole, hitting a monster drive down the 18th and then a fine second shot to enjoy the luxury of two putts for the crown.
Wallace, who had found himself three over through eight holes of his opening round, said: “I’m ecstatic. I just played some amazing golf.
“Ever since being three over through eight at the start of the week, I’ve played some of the best golf of my life and to do it in that style at the end there capped it off.”
Wallace shared the lead with in-form home favourite Shubhankar Sharma heading into the final round, but the Englishman stormed in front with five birdies and a bogey to card 68.
Sharma, who broke the course round on Friday with an eight-under-par 64, saw his challenge disintegrate as he registered a three-over 75 which contained three double-bogeys, three further dropped shots and six birdies.
Johnston produced a blemish-free 66 in the final round to surge within touching distance of victory, while American Sihwan Kim (68) finished in sole third on eight under.