A strike after five minutes from 18-year-old Tyler Dibling put the hosts ahead as Ipswich failed to clear the ball in their box and it was worked to Dibling, who beat a defender and coolly shot into the back of the net from 10 metres for his first goal in senior football.
Southampton also hit the post but the better chances fell to the visitors, who were wasteful with their finishing and found Aaron Ramsdale in good form in the home goal.
They managed to get an equaliser when a corner was cleared to Morsy deep into injury time and his fierce drive took a deflection off a defender and flew into the top corner.
Ipswich rise to 16th place, having taken three points from a possible 15 at the start of the season, while Southampton are in 18th with their first point of the campaign.
While there will be relief all round, Ipswich remain winless, the first time they have failed to pick up a victory in their first five games of the season since 1984.
Southampton striker Ben Brereton Diaz extended his unfortunate record of Premier League games without a win to 19.
Ipswich’s tally of six shots on target was double that of their hosts and they will feel their point was deserved.
Sammie Szmodics wasted an excellent early chance at the back post and then headed straight at Ramsdale from eight metres.
Southampton came close to doubling their advantage against the run of play when Cameron Archer beat two defenders and rounded the goalkeeper but fired his shot on to the post.
Archer had another chance to score early in the second period but shot too close to the goalkeeper, and Southampton’s failure to kill the contest when they had the chance left the door open for Ipswich to claim a point.
