The Championship new boys have signed the 19-year-old from Colchester for an undisclosed fee.
The transfer was made official on Monday as Poku penned a four-year contract.
“I am really excited about this signing. He is a young player I have watched for the last 18 months. We enquired about him last season and we have continued to monitor his progress,” said Darren Ferguson Peterborough boss.
“He had a bit of a stop-start season last year, which can happen with young ones. He can play as a 10 and he can play wide right or even wide left at a push. He is a talented boy and I believe he has real potential. It is a really good signing for the football club.”
Poku made 75 appearances for Colchester, scoring six goals. His last two goals for the club came in a 3-0 win over Carlisle United in March.
Poku said: “I am really happy to be here, it has taken a little while because I have actually had Covid, but it is all good, things like that can happen, I am really looking forward to getting started. This club have shown they can develop younger players and I am excited to get going.
Black Stars coach Charles Akonnor in March named a squad that included Poku – his debut call-up.
And despite the criticisms, because Poku was playing in the fourth-tier league, he played 11 minutes in Ghana’s final Africa Cup of Nations qualifying match against Sao Tome and Principe.