The 24-year-old joins the Tractor Boys for an undisclosed fee, although Accrington have confirmed that the deal is worth more than £1m, making Jackson a new record sale for the newly-promoted League One club.
Jackson has penned a three-year contract at Portman Road with the option of a further 12-month extension on top of that, potentially keeping him at the club until 2022.
"This time about 13 months ago, I didn't know where I would be, whether I would be playing first team football at all, now I have a medal to look at and it has easily been the best year of my life," Jackson told Accrington's official website upon his departure.
"It has been hard few weeks, there were a few offers from the Championship, I could have stayed in League One and I would have enjoyed it but it's a good opportunity at a club where I really believe I have the opportunity to play games and score goals in the Championship. It is a dream come true for me from where I was 13 months ago.
"While I have my dream move, it is also quite sad. I loved every minute of my time at Accrington, I had my own song which was great and from day one every Accrington fan has been positive with me. I might not have had the best games all the time but the fans never got on my back, I never got an ounce of stick. There may not be 5,000 there every week but they were unbelievable and were class with me.
"I have a lot to thank Accrington for and I think I also helped them. I made memories I will never forget, I got on with everyone at the club and made some friends. It's like a big family at Accrington, it's a great atmosphere and it's so easy to settle in, everyone made it so easy for me."
Jackson, who scored 16 goals to help Accrington to the League Two title last season, becomes the ninth summer signing for new Ipswich boss Paul Hurst.