Woods was aggressive from the outset on day four as he bid to make up lost ground on the leaders.
His challenge was dented by a dropped shot at the fifth, before he bogeyed the seventh in a difficult front nine.
Successive birdies at nine and 10 breathed new life into his chase but time and holes were fast running out.
Two more birdies followed at 13 and 15 for Woods, but a short putt on a birdie attempt at 16 effectively ended his chances of catching the top three.
Despite finishing with a two-under round of 70, the 37-year-old could not pull off a dramatic comeback and had to settle for a place in the top 10.