Woods, 51, will take over the post on January 1.
"This change in leadership is consistent with the board of directors' succession plan developed years in advance," the Texas-based company said in a statement yesterday, adding this "demonstrates the strength of the management development system".
Woods will take over the post on January 1. He has been with Exxon for nearly a quarter-century.
He rose through Exxon's refinery and chemicals businesses and emerged as Tillerson's likely successor a year ago, when he was named president and joined the board of directors.
He was president of the Exxon Mobil Refining and Supply Company for two years before being named senior vice president for the corporation and a member of its management committee in 2014.
64-year-old Tillerson has been with the oil giant for more than 41 years.
Tillerson was expected to step down in early 2017, when he will hit Exxon's mandatory retirement age of 65. His selection by Trump for the nation's top diplomatic job - subject to Senate confirmation.
"The company doesn't change, but obviously it tries to adapt its strategy to the business environment," he said.
Exxon is sitting on USD 5 billion in cash and has a stock-market value of USD 375 billion. Gheit said that Woods might try to take advantage of lower values in the oil industry to grow by making one or two major acquisitions.
In a statement, the Exxon board praised Tillerson and said, "We know that his service to the nation as secretary of state will be equally successful and distinguished.
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