Paul Anderson gave the visitors a 12th-minute lead but it was quickly cancelled out by Michail Antonio's equaliser.
The result leaves Wednesday still waiting to record their first win of the season and Ipswich without a victory on their travels.
Both sides hit the woodwork in the match following fine, long-range efforts from Ipswich midfielder Cole Skuse and Wednesday substitute Jermaine Johnson.
Ipswich striker Daryl Murphy had an early shot blocked while Wednesday's Atdhe Nuhiu was slow to react when he found himself in a good position and his shot on the turn was saved by Dean Gerken.
Town caught Wednesday on the break for the opening goal following an attack from the home side which came to nothing.
David McGoldrick broke free on the left wing and his cross found the unmarked Anderson, who fired past Chris Kirkland.
Wednesday made a swift response, getting back on terms two minutes later as Reda Johnson's cross was headed back by Nuhiu at the far post and Antonio's header beat Gerken.
Skuse went close to putting the visitors back in front in the 36th minute when his firm 20-yard drive beat Kirkland but the ball smacked against the keeper's right-hand post and bounced away.
Wednesday's Matty Fryatt missed a great chance to score at the end of the half when he was free in the middle of the area and shot straight at Gerken who turned the ball away for a corner.
Fryatt will no doubt feel that he should have done better with the opportunity.
In the second half, Jeremy Helan, under pressure, headed wide for the home side soon after the re-start.
Murphy had a shot saved by Kirkland the keeper was called into action again to keep out Anderson's angled shot at his near post.
McGoldrick was the next to try his luck for the visitors but his long-range effort was off-target.
Johnson, on as a substitute for Wednesday, went close to scoring spectacular goal when his 25-yard strike thumped Gerken's left-hand post.
In the closing stages of the match, neither side really threatened to snatch a winner and a draw, on reflection, was probably a fair result.
Wednesday remain close to the foot of the Championship table after collecting just five points from their opening 10 games of the season.