The visitors took the lead 10 minutes into the second half when Peter Vincenti was left unmarked and headed in from close range at the back post.
Jamille Matt equalised late on when he rose to nod in Bobby Grant's corner.
Vincenti won a late spot kick, going down after a challenge from Eggert Jonsson, but Maxwell pulled off a superb low save to deny Ian Henderson.
Rochdale manager Keith Hill told BBC Radio Manchester:
Media playback is not supported on this device
"We've drawn a game we could have won but you've got to respect the opposition. We've had another very good performance.
"You've got to admire the bravery of Hendo [Ian Henderson]. He's missed a few penalties and also scored a few very important ones.
"I thought it was a good save.
"The players have got to be a bit disappointed because obviously they're ambitious.
"But when they actually assess and get over the emotions they've been through, they will realise that we're a good team and we've got a very good point here."
Fleetwood Town goalkeeper Chris Maxwell saved a stoppage-time penalty to secure his side a draw against Rochdale.