After Robins had steered his new side to their first win in midweek at Derby County, it was a first home point for the Tykes - with 34-year-old Moore at the heart of the home side's impressive resistance.
The new manager was given a noisy standing ovation before kick-off and the home side made a promising start against the Welsh outfit, who are also languishing in the lower reaches of the Championship.
It was the visitors that flashed an early warning when, after 11 minutes, Mark Gower drove in a shot from 20 yards but it was off-target.
At the other end Iain Hume's corner was a fraction short of connecting with Moore's head.
And when Swansea broke again, Barnsley were caught out but keeper David Preece blocked Gorka Pintado's shot.
There was a scare for the home side when Hume - recently returned from a life-threatening skull fracture, went down clutching his head sparking fierce protests from the home side.
The Canada international had to leave the pitch for stitches to stem heavy bleeding from a head wound but he made a quick return.
The home side should have been in front after 26 minutes when Andy Gray beat the offside trap but his shot under pressure was blocked.
And when Robn Kozluk floated a free-kick into the Swansea area, Moore's header was just wide before Swansea's Cedric van der Gun glanced a super header narrowly off-target from Andrea Orlandi's corner.
After the break a mistake by Jacob Butterfield and Hugo Roberto Colace let in Swansea, but keeper Preece made a vital stop to deny Nathan Dyer.
Another opportunity came Barnsley's way when Franca Anderson Silva's precise pass sent in Hume who shot over when Andy Gray was unmarked.
And Swansea sub Craig Beattie wasted a late chance when he fired wide.