An activist investor is likely to pick up a third seat on the board of Exxon XOM 0.79% Mobil Corp., giving it additional leverage to press the oil giant to address investor discontent about diminished profits and its fossil-fuel focused strategy amid concerns about climate change. Exxon said Wednesday that an updated vote count showed shareholders backed a third nominee of Engine No. 1, an upstart hedge fund that had already won two board seats at Exxon’s annual shareholder meeting last week. The final vote hasn’t been certified, Exxon said, and could take days or weeks to be finalized, according to people familiar with the matter.

Engine No. 1, which owns a tiny fraction of Exxon’s stock, had sought four seats on the board and argued the Texas oil giant should commit to carbon neutrality, effectively bringing its emissions to zero—both from the company and its products—by 2050, as some peers have. If the preliminary voting results hold, it will control a quarter of Exxon’s 12-person board.

The vote culminated one of the most expensive proxy fights ever. It puts new pressure on Exxon Chief Executive Darren Woods, who personally campaigned against Engine No. 1 and could complicate his plans to maintain Exxon as the largest Western oil producer. Mr. Woods was re-elected to the board along with eight of Exxon’s candidates.

“We look forward to working with all of our directors to build on the progress we’ve made to grow long-term shareholder value and succeed in a lower-carbon future,” Mr. Woods said in a statement. “We thank all shareholders for their engagement and participation, and their ongoing support for our company.” Though Engine No. 1 only owned 0.02% of Exxon’s stock, it was able to capitalize on investors’ concerns about Exxon’s recent performance and climate change strategy. Exxon posted a $22 billion loss last year, its largest on record, after the pandemic crushed fuel demand and upended what turned out to be an ill-timed plan by Mr. Woods to substantially increase spending to boost oil and gas production.

Many of the world’s biggest investment firms helped elect the fund’s directors, including BlackRock Inc., State Street Corp. and Vanguard Group, who said publicly they wielded votes for investors in favor of at least two dissident directors.

BlackRock voted for three directors proposed by Engine No. 1 and said last week it believes Exxon and its board need to further assess the possibility that demand for fossil fuels may decline rapidly in the coming decades. Vanguard voted for two directors proposed by the activist and said it had expressed concerns to Exxon for years about the independence of Exxon’s board and its directors’ lack of energy sector expertise. “And for years, we did not witness sufficient progress on either front,” Vanguard said last week.