The 21-year-old twice dragged shots wide when he was through on goal and also headed a corner over the bar from point-blank range.
Home goalkeeper Barry Roche also made a good save from the former Leicester City trainee and Barnsley wasted another promising opening when Paul Hayes shot wide in the 90th minute.
Chesterfield had only one clear chance, which Caleb Folan put inches wide of a post in the 81st minute, despite having long periods of possession.
The first half was dominated by the Spireites but the final ball was not good enough to open up a solid Barnsley defence.
Derek Niven had a header saved by Nick Colgan but there were few other goalmouth incidents before the interval.
Barnsley soaked up the pressure and started to catch Chesterfield on the break, and Wright should have done better in the 51st minute.
The striker got away on the Chesterfield right but his low shot was wide of Roche's left post.
He had an even better chance 20 minutes later when he was clear through the middle, but he went for power and Roche pushed the ball away diving to his right.
It just would not fall for Wright, whose worst miss came in the 83rd minute when he headed over the crossbar when unmarked.
Both teams showed signs of tiredness but Barnsley had more chances to take the points in the closing minutes, only for their finishing to let them down.
Substitute Hayes found space on the Chesterfield right in the 90th minute but he put his shot into the 1,800 Barnsley fans behind the goal.