Michael Chopra was denied a penalty just before Jay Emmanuel-Thomas fired into
the side-netting as the hosts started at a frantic pace in the early stages.
The away side then threatened after Cotterill and Gray worked the ball to Davies down the right. His cross looked to have been deflected into the net but Lee-Barrett scrambled back to prevent an own goal from Edwards.
On 20 minutes, Drury hit a shot just over the cross-bar from 25 yards.
Ipswich were on top in the game and saw another effort go close when Daryl Murphy's acrobatic attempt from close-range went just over the top.
Lee-Barrett nearly gifted Barnsley a goal from Korey Smith's ambitious 30-yard effort, the goalkeeper let Smith's effort squirm under his body but recovered to stop the ball right on the line.
Higginbotham played a well-threaded pass that almost found Davies in on goal, but Lee-Barrett was alert to the danger and stopped the striker excellently a minute before the break.
The sides went in at half time with the scores still level.
Three minutes after the restart, Ipswich took the lead through a Daryl Murphy goal.
Barnsley looked unlikely to threaten the home side's goal as Ipswich dominated the second half.
The Tractor Boys saw out the game to claim all three points thanks to Murphy's early second-half goal.