The 66-year-old signed an 18-month contract with the Midlands club, who sit 19th in the Premier League with seven points after 13 games.
West Brom will be the eighth club Allardyce has managed in the Premier League, after taking charge of Bolton Wanderers, Newcastle, Blackburn, West Ham United, Sunderland, Crystal Palace and most recently Everton.
Allardyce's first match will be against Aston Villa on Sunday and the club's sporting and technical director, Luke Dowling, said he was sure they had the right man for the job.
"In Sam we have a man who has a proven Premier League pedigree with a track record of improving every club he has managed," he said.
"We believe and, more importantly, Sam believes we have a group of players that have the quality needed to give the club its best chance of Premier League survival."
Bilic took over at West Brom in the summer of 2019 and won automatic promotion to the Premier League.
West Brom drew 1-1 at Manchester City on Tuesday but the decision had apparently already been made and Bilic was told the news on Wednesday morning.
"Albion would like to thank Slaven and his coaching staff for their efforts in achieving promotion last season and wishes them all well in the future," West Brom said in a statement.
Manchester United manager Ole Gunnar Solskjaer said he felt sorry for Bilic.
"Well I think these times in a pandemic and what has happened has made everyone sit back and think," he said.
"I'm very sad to hear Slaven has lost his job. He did a great job getting them promotion. I've known him for years and I'm sure he'll be back soon.
"Hopefully more and more clubs will think long term. It's a short term business and you need short term results, but continuity is the key to success and patience."
