LONDON—Prime Minister Theresa May won formal approval from the House of Commons on Wednesday to hold an early election in June, supported by an opposition Labour Party that could be facing one of its worst national defeats in decades.

The 522-13 vote puts the Labour Party in an uncomfortable position as it grapples with waning support from its traditional working-class base and low approval ratings in comparison with Mrs. May’s Conservatives. Pollsters and analysts say Mrs. May, in the June 8 election, could increase her majority...