This will be the third loan spell that the 19-year-old has had away from the Hawthorns, having previously spent time with Northampton Town and Brentford.
Discussing Berahino's arrival, Posh manager Darren Ferguson admitted that the centre-forward is a player that he has admired for a long time.
"Saido is a player that we have watched for a considerable amount of time. He has been out on loan at a number of clubs and he is a good player. Any manager would say the same, you need as many good players in the building as possible," he told the club's official website.
"He is a striker with good movement and he has a good goalscoring record. We are fortunate that West Brom have allowed him to join us.
"I tried to get him last week as he was involved in the first team, but they feel it is the right time for him to come out on loan."
Berahino is expected to be in the squad for tomorrow's encounter against Barnsley.