Boeing Co. ’s board must face a shareholder lawsuit claiming directors failed to properly monitor safety matters before two 737 MAX jets crashed, and then “publicly lied” about their oversight of management, according to a Delaware judge’s ruling.

In an opinion issued Tuesday, Vice Chancellor Morgan Zurn of Delaware’s Court of Chancery let stand one of the lawsuit’s claims while dismissing others, moving it past an initial hurdle.

“Rather than prioritizing safety, defendants lent their oversight authority to Boeing’s agenda of rapid production and profit maximization,” she said. The judge’s opinion faulted the board for lacking a safety-reporting process and then “turning a blind eye” once a problem emerged.

Boeing, which had sought to dismiss the case before it advanced, can appeal the judge’s ruling. A spokesman said the company would review the judge’s opinion closely as it considered its next steps. “We are disappointed in the court’s decision to allow the plaintiffs’ case to proceed past this preliminary stage of litigation,” he said.

Vice Chancellor Zurn’s 102-page opinion also said Boeing’s board “publicly lied about if and how it monitored the 737 MAX’s safety.” She cited then-Lead Director David Calhoun’s statements to the news reporters, including that the board considered grounding the plane before the second crash. “Each of Calhoun’s public representations was knowingly false,” she wrote.

Mr. Calhoun is now Boeing’s chief executive.

Officials inspected an engine recovered from a Lion Air 737 MAX that crashed in Indonesia in 2018.

Photo: Achmad Ibrahim/Associated Press

The company has previously said the lawsuit, which was brought by New York and Colorado public pension funds, presented a misleading picture of the board’s actions and distorted accounts of Mr. Calhoun’s interviews. Boeing has said its board engaged in robust safety oversight that included extensive reviews of aircraft engineering and manufacturing. Since the crashes, the board has set up a committee to specifically oversee aerospace safety.

In her opinion, Vice Chancellor Zurn faulted the board for not questioning Boeing management’s assurances after The Wall Street Journal reported in November 2018 that a new flight-control system known as MCAS was suspected of playing a significant role in the crash of a Lion Air 737 MAX in Indonesia.

“The Lion Air Crash was a red flag about MCAS that the board should have heeded but instead ignored,” she wrote.

At the time, accident investigators were also focusing on missteps by the airline’s crew and maintenance workers. Boeing and U.S. air-safety regulators reminded pilots of an emergency procedure that could disable MCAS while engineers worked on additional safeguards for the flight-control system. A second jet crashed before the system could be fixed. The accidents took a total of 346 lives.

Surviving a motion to dismiss in such cases targeting company boards, while historically rare in Delaware, is becoming more frequent after a state supreme court ruling there two years ago, said Elizabeth Pollman, a corporate law professor at the University of Pennsylvania.

While the Boeing case hasn’t yet played out, the judge’s opinion Tuesday is likely to have broader influence in other corporate boardrooms as company leaders consider what director-level safeguards they might need to avoid major risks, she said. “They read these opinions for guidance,” said Ms. Pollman, who is co-director of Penn Law’s Institute for Law & Economics.

Plaintiffs in the Delaware case have been able to gain access to more than 630,000 pages of internal board documents, including emails and meeting minutes, under the state’s law. The judge’s ruling is expected to allow plaintiffs to pursue traditional discovery that could include additional requests for documents and depositions as the litigation continues.

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The Boeing board lawsuit is different from a securities-fraud lawsuit through which plaintiffs themselves might potentially recoup investment losses. Known as a shareholder derivative complaint, the legal action seeks to hold company officials responsible for their alleged missteps and, if ultimately successful, could result in defendants paying monetary damages to the corporation and prompt internal governance changes. In such cases, plaintiffs can sue current and former directors on behalf of the company, if a court agrees that directors wouldn’t pursue the action themselves.

Law professors said such cases tend to be settled before going to trial, with defendants’ insurers generally making any potential monetary payments.

Last year, Wells Fargo & Co. reached a settlement with plaintiffs who filed a similar legal action related to the scandal over the bank’s sales tactics, according to court documents in the case. The agreement included $240 million paid by defendants’ insurers to the corporation, as well as governance changes and clawbacks, according to the court documents.

A Wells Fargo spokeswoman pointed to a bank securities filings last year that noted the company would receive the $240 million and pay plaintiffs’ attorneys’ fees.

A firm that represented plaintiffs in the Wells Fargo case was Lieff Cabraser Heimann & Bernstein LLP, which also represents the plaintiffs in the Boeing case.

Write to Andrew Tangel at Andrew.Tangel@wsj.com