Barnsley thought that they had equalised towards the end of the first half when they had the ball in the back of the net, but the referee disallowed it for a foul on Scott Loach.
Luke Chambers almost doubled Ipswich's lead at the start of the second half with a headed effort, but Alnwick just kept it out.
Barnsley came within inches of finding the equaliser ten minutes later when Chris Dagnall found space to shoot, but his effort hit the inside of the post.
They were level with just over 15 minutes of the match left, however, as Stephen Dawson scored with a fine volley.
Neither side could find a winner in the closing stages as they had to settle for a point apiece.