Sammie Szmodics George Hirst had the visitors 2-0 up; before YoaneWissa, an own goal and a Mbeumo penalty turned things around.
Harry Clarke was sent off for Ipswich, but Liam Delap then drew scores level at 3-3 before Mbuemo found a winner in injury-time.
Mbeumo later said: "I know this type of cross can be dangerous, either a flick or no one touches it, and the keeper has doubt. In the end it was a goal and it was a really important win today.
"They gave it their all in defence (with ten men), so it was really hard to find the gaps. It is not always easy but I think we showed big character today.
"They defended really well and they were really good on the counter attack as well. We knew we had to stay in the game no matter what and that is what we did. We always try to stay together, stick to the plan, and we know we have the quality to get back and that is what we tried to show.
He addedt: "Maybe (there was) a lack of focus, but we need to enjoy this kind of win.
"It is really important to stay unbeaten at home and make it tough for the opponents to come here. We are going to try to continue to do it."