Conor Washington gave the visitors a first-half lead, but a late Barnsley corner saw Berry head home in the final minute of stoppage time.
Peterborough took the lead in the 16th minute through Washington, with the midfielder pouncing from close range after Jack Cowgill had failed to clear a long ball into the penalty area.
Barnsley had their chances to get back in the game in the first 45 minutes, with Conor Hourihane failing to capitalise on an open goal from edge of the box, while Berry dragged a shot wide of Ben Alnwick’s goal.
The hosts started the second half well, with Sam Winnall going close on two occasions early on.
In the 76th minute, Kgosi Ntlhe was sent off after receiving a second yellow card for a crunching tackle on Josh Scowen.
Barnsley left it late to equalise, with Hourihane’s inswinging corner finding the head of Berry in the penalty box and he secured a point for the hosts.