LONDON—British Prime Minister Theresa May won a key vote on Brexit legislation early Tuesday, but she faces tough battles ahead in getting Parliament to support her vision for how the U.K. should exit the European Union after more than four decades.

Lawmakers voted 326-290 in favor of a bill designed to transpose more than 10,000 EU laws on to the U.K. statute book. The bill would come into effect on March 29, 2019, the day the U.K. is scheduled to leave the bloc and aims to prevent a legal vacuum once Britain leaves the EU.

However, critics argue the bill hands too much power to the prime minister and her cabinet because it allows them to alter laws without parliamentary approval.

The bill’s difficult journey through the early stages of parliamentary scrutiny—normally a formality—signals further hurdles along the line for Mrs. May, who lost her party’s majority in an election gamble earlier this summer. While negotiations with the EU over Britain’s departure have reached an impasse over issues such as how much the U.K. owes the bloc as part of its divorce, a bigger issue for Mrs. May could be getting a divided Parliament and country behind her negotiating aims.

Mrs. May said the bill gives “certainty and clarity” ahead of Brexit. “Although there is more to do, this decision means we can move on with negotiations with solid foundations and we continue to encourage MPs from all parts of the U.K. to work together in support of this vital piece of legislation,” she said.

The vote on the bill is just one step in a longer legislative process. While some lawmakers who supported staying in the EU say they will vote in favor of the bill, they will seek to attach amendments at a later stage that restrict the government’s authority to make substantial changes to U.K. law without parliamentary approval, such as watering down to laws protecting workers’ rights or environmental standards.

Keir Starmer, the opposition Labour Party’s Brexit spokesman, said in an interview with The Wall Street Journal that the government’s attempt to weaken the role of lawmakers in the Brexit process had set the stage for a lengthy standoff between Mrs. May’s cabinet and the rest of Parliament.

“This is only the beginning of quite a turbulent two years,” Mr. Starmer said ahead of the vote. “There are a number of shared concerns across the house about the nature of this bill.” He said after the vote that Labour would seek to remove the worst aspects of the bill as it passes through Parliament, but its flaws are so fundamental that it was hard to see how the bill could be made fit for purpose.

Brexit Secretary David Davis warned the U.K. would descend into chaos if the bill isn’t approved.

“The British people did not vote for confusion and neither should Parliament,” Mr. Davis said ahead of the vote.

Mr. Starmer said Labour isn’t voting against Brexit, but against the principle that ministers should have the power to modify elements of EU law once they are incorporated into U.K. law. These powers are known as Henry VIII clauses, after a 16th-century statute that gave the king power to legislate by proclamation.

“Even if you’re a Labour MP that campaigned and voted for leaving the EU, you still think it’s right that Parliament has a say over what the withdrawal looks like,” Mr. Starmer said. “Whether we’re leaving is a closed question, but how we’re leaving isn’t.”

The Labour Party has increased pressure on the Conservatives to pursue a closer relationship with the EU than Mrs. May has outlined. Mr. Starmer said the U.K. shouldn’t rule out staying in the EU’s customs union indefinitely if trade deals forged outside it won’t make Britain better off.

The bill’s scope highlights the complexity of leaving the EU, a process that has absorbed most of Parliament’s time. A report by the think tank Institute for Government published Monday said introducing customs checks after Brexit could cost more than £4 billion ($5.3 billion) a year and that the task would require changes across more than 30 government departments and local