Posh broke the deadlock just past the mid-point of the opening period when Danny Swanson notched a delightful first goal for the club.
Charlton skipper Johnnie Jackson levelled when bursting onto a Callum Harriott pass and firing across goalkeeper Bobby Olejnik in the 55th minute, and that was swiftly followed by a second from hitman Danny Haynes, who pounced after Olejnik had clawed out a Michael Morrison header.
Peterborough refused to roll over and Bostwick restored parity when his bouncing long-range blast deceived the diving Button, but the Charlton keeper redeemed himself when blocking a stoppage time header from Tommy Rowe as the hosts searched for a winner to no avail.
"I think a point apiece was about right overall," said Charlton boss Chris Powell. "We started well only to find ourselves behind, but then responded very well in the second half to get in front.
"We were well aware of the fact that Michael Bostwick has a good strike on him. It's something we spoke about beforehand but unfortunately we couldn't stop him unleashing one to get Peterborough level.
"Both teams had chances and both could have won it, but I'll settle for a point after a couple of poor results lately.
"We're only six points clear of the relegation zone so we can't afford to be complacent. That sort of gap can be wiped out very quickly so and we must continue to work hard."