The Ireland international, 27, has signed a four-year deal at Portman Road and becomes their 10th signing since they returned to the Premier League.
The Tractor Boys had been looking for another forward despite spending almost £100m this summer, after winger Wes Burns suffered a hamstring injury in the 2-0 opening day defeat by Liverpool.
“I’m very excited and feel lucky to have this opportunity to play in the Premier League again,” said Ogbene.
“I’ve had very positive chats with the manager [Kieran McKenna] and for someone to believe in me, that I could play at this level again, means a lot to me and my family.
“I know what he wants from me and the role I will play and I’m here to be a positive addition to the squad as we look to achieve our goals.”
Ogbene, who has also played for Cork, Limerick and Brentford, joined Luton from Rotherham on a free transfer last summer and impressed in the Premier League, despite the Hatters’ relegation.
He scored five goals in 39 games in all competitions and was the second fastest top-flight player last season, after Tottenham’s Micky van de Ven, having clocked 36.93kph against Fulham last September.
Ipswich, who travel to AFC Wimbledon in the Carabao Cup on Wednesday, also remain in talks with Chelsea for striker Armando Broja ahead of Friday’s transfer deadline.
McKenna said: “We’ve spoken to him, I think that’s well known. There’s ongoing discussions with Chelsea but I’ve not been privy to them over the last 24 hours. We’ll see if it’s the right thing for the club and the player.”