Both goals came from set-pieces as the Owls cancelled out Dean Freeman's early header by levelling direct from a corner.
The two sides were evenly matched in this tough-tackling South Yorkshire derby, and Wednesday edged the latter part of the second half.
The Reds' youngsters had Chadderton in goal to thank for keeping the score at 1-1 late on, moments after Dawes had cracked the crossbar from long range.
Barnsley's goal came after five minutes, when Alexander-Salmon's corner was powered in off the bar by the head of Freeman.
But with ten minutes to play, the Owls levelled as an in-swinging corner caught Chadderton out and found the top corner.
Reds: Chadderton, Smith, Cowgill, Freeman, Holgate, Bree, Biggins, Alexander-Salmon, Maris (Templeton 75), Nana, Pilkington.