Woods, 51, 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.
The company announced Wednesday that Woods will take over the top spot on Jan. 1.
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 — pushed up the timing.
Many analysts expect an orderly transition at Exxon, with few significant changes in policy or leadership style under Woods.
Because of Exxon's huge size, strategic decisions are made with a long time frame in view. The Irving, Texas-based company is known for taking a cautious, methodical approach to decision-making, and changes in direction usually happen only slowly.
Woods "has been groomed for a couple years to be the replacement," said Stewart Glickman, an analyst for CFRA Research. "He is a 24-year (Exxon) veteran, and I think he probably understands the culture pretty well. I don't think there is much of a hitch for Exxon."
Brian Youngberg of Edward Jones said given Exxon's size "some tweaks may occur, but I doubt anything substantial" —especially because the company is already an industry leader in production, refining, chemicals, and selling finished goods like gasoline.
Fadel Gheit, an analyst with Oppenheimer & Co. who has followed Exxon for 30 years, said the new CEO would be "old wine in a new bottle."
"The company doesn't change, but obviously it tries to adapt its strategy to the business environment," he said.
Exxon is sitting on $5 billion in cash and has a stock-market value of $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."
Tillerson replaced longtime CEO Lee Raymond in 2006 and led the company during one of the most turbulent periods in its history, which dates to the Standard Oil Co. founded by John D. Rockefeller. The financial crisis of 2008 led to a collapse in oil prices. Crude recovered, only to plunge again starting in mid-2014 because of worldwide oversupply.
As CEO, Tillerson signed a multi-billion-dollar deal to explore for oil in Russia's Arctic, only to see the project slowed when the U.S. and European allies slapped sanctions on Russia for annexing Crimea from Ukraine. Analysts said Tillerson overpaid for XTO Energy in 2009. The deal made Exxon the biggest natural gas producer in the U.S. but hasn't lived up to expectations because of continuing low gas prices.
Exxon earned a record $45.2 billion in 2008 and nearly matched that in 2012. Since then, however, profit has fallen, dropping to $16.2 billion last year — a whopping sum but Exxon's smallest gain since 2002.
The company lost its coveted AAA credit rating earlier this year. Its stock price has risen 16 percent in 2016, but rival Chevron's shares have gained nearly twice as much.
On Wednesday, Exxon shares fell $2, or 2.2 percent, to close at $90.58.