Anheuser-Busch InBev NV finally won over SABMiller PLC’s board with a $104.2 billion takeover bid to create a brewer that would dominate much of the world’s beer market, but only after weeks of behind-the-scenes talks that erupted into a testy public standoff.

SABMiller, the maker of Miller Lite and Pilsner Urquell, said Tuesday its board would recommend its shareholders approve AB InBev’s latest proposal of £44 a share—a 50% premium to SABMiller’s Sept. 14 share price, the day before media speculation about a potential deal...