The Portman Road side, without a victory in nine games, had taken a 3-1 lead before United burst into life, Darius Henderson finding the target in the 76th minute with Morgan heading home deep into extra time.
Struggling Ipswich were in trouble in the 9th minute after Ched Evans fired in a low strike which was blocked by centre back Gareth McAuley, but as the ball spun in to the air, Henderson bundled the ball over the line with a close-range header.
Midfielder Stephen Quinn, facing his older brother Alan, almost grabbed a second with a stinging right foot strike after cutting in from the left flank.
Ipswich were back on level terms in the 24th minute after the United back four failed to clear midfielder Carlos Edwards' low cross which Jon Walters was the first to reach, sticking out a foot to steer the ball beyond the reach of Mark Bunn.
Ipswich added a second in the 33rd minute with the United keeper at fault. Midfielder Grant Leadbitter started the move and finished it, linking with Walters before collecting the return pass and hitting a low shot from 30 yards. Bunn had a clear view as he dived to his left as the ball spun beyond his reach.
Former United striker Jon Stead replaced the ineffective Tamas Priskin on 60 minutes and immediately carved out a half-chance; blasting in a low strike which was smothered on the line by Bunn.
Richard Wright showed his class minutes later, conjuring up a one-handed reflex save to palm away an Andrew Davies header after Ipswich failed to cut out David Cotterill's precise cross.
Wright pulled off another less spectacular stop diverting a Stephen Quinn shot with his outstretched leg before McAuley added a third on 76 minutes jumping unmarked to reach Leadbitter's corner and power his downward header beyond Bunn.
Henderson grabbed his second of the night 14 minutes from time with Morgan heading in substitute Glen Little's corner with time running scarce.