The Tractor Boys are floundering at the wrong end of the Championship, but looked to be on their way to a first win in six games when Aaron Cresswell fired in a free-kick in the opening 10 minutes, which Barnsley goalkeeper Ben Alnwick should have kept out.
The hosts struggled for a response for much of the game, though they came close when Chris Dagnall fired against the inside of the post.
They went one better with 15 minutes remaining as Dawson thundered in an exquisite volley with the outside of his point to deliver a worthy point for Keith Hill's side.
The draw sees Town entrenched into the bottom three of the early-season table and a winter of discontent looks on the horizon, while Barnsley have lost just one of their last five games and are sitting comfortably in mid-table.
Jewell earned the backing of chief executive Simon Clegg during the week amid rumours of Harry Redknapp replacing him at Portman Road after a disappointing start to the campaign.
It certainly does not seem as if Jewell has the backing of the travelling Town supporters as they unveiled a 'Jewell Out' banner, but the beleaguered boss' hopes of remaining in the job received a boost with the game just six minutes old.
Andy Drury laid a free-kick off to and Cresswell fired in a 20-yard free-kick that Alnwick failed to keep out and the ball went into the bottom corner.
Barnsley came into the game on the back of the superb 5-0 win at Birmingham last week, but they failed to pick up from where they left off as they started slowly.
Indeed Town could have been two up midway through the first half as they broke quickly and Lee Martin fired over from a good position.
Craig Davies, who scored four at St Andrew's last week, tried to haul his side level on the half-hour, but he dragged his shot wide after a neat passing move.
The Oakwell fans thought their side had forced a leveller shortly before the break, when Stephen Foster headed in, but he was adjudged to have fouled Scott Loach.
Things almost got even worse for the Tykes six minutes after the restart as the visitors should have doubled their lead.
Luke Chambers' climbed highest from a corner and his header was superbly parried by Alnwick and Danny Higginbotham prodded the rebound just wide.
Martin had another effort go just wide before Ipswich's goal lived a charmed life as Barnsley came within inches of equalising.
Dagnall found some space and fired in a low effort that hit the inside of the post and rolled across the line before rolling clear.
But the hosts were not be denied for much longer as Barnsley levelled with 15 minutes to go and it was a fine finish.
A sweeping move found Jacob Mellis and he sent a fine ball over for Dawson, who finished with a sumptuous volley with the outside of his boot.
That gave Hill's men the impetus to go for the win and Loach was forced into a fine save as Ipswich were left hanging on.