The 27-year-old Republic of Ireland international, who scored four times in 30 appearances for the Hatters in last season's English top flight, has signed a four-year contract.
"I'm very excited and feel lucky to have this opportunity to play in the Premier League again," Ogbene told Ipswich's website.
"It's an opportunity I had to grab with both hands and I can't wait to be involved as soon as possible."
Ogbene started his career in Ireland with Cork City and Limerick before spells at Brentford, Rotherham and Luton.