With tournament host Arnold Palmer watching on, Day got up-and-down from a greenside bunker at the final hole, sinking a four-foot par putt after earlier holing a clutch 12-foot birdie at the par-three 17th.
"I just ground it out today," the Australian said in a greenside interview after carding a closing 70 on the Bay Hill course in Orlando.
He finished at 17-under-par 271, while American Kevin Chappell (69) claimed second place on 16-under after bogeying the last.
Day, 28, collects $1.134 million for his eighth victory on tour. He also rises from third to second in the world rankings.
It is the third time in four events an Australian has won on the U.S. circuit, with Adam Scott having triumphed at the Honda Classic and WGC-Cadillac Championship.
Day revealed he had received a text message on Sunday morning from eight-times Bay Hill winner Woods, who is out indefinitely while recuperating from back surgery.
The long-hitting Australian grew up wanting to follow in the footsteps of Woods, who is 12 years older.
But Day did not play like the 14-times major champion early in the final round, running up three bogeys in the first six holes to open the door to his rivals.
It then became a three-man race late in the round between Day, Chappell and American Troy Merritt.
Chappell, playing ahead, had the lead with one hole left, but dropped a shot at the par-four 18th after a poor drive finished in punishing rough and gave him little choice but to lay up.
That opened the door for Day, who also pushed his drive at the last, but drew a decent lie and was able to clear the pond in front of the green and advance his ball into a bunker.
Merritt, meanwhile, could have forced a playoff with a birdie at the last, but instead found a watery grave with his approach shot, ran up a double-bogey and tied for third with Swede Henrik Stenson, three shots off the pace.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Mark Lamport-Stokes)
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