Crowe was given his marching orders for a second bookable offence, something his manager Stuart Gray described as 'a tired tackle'.
Peterborough led through George Boyd's fierce 16-yard shot after the cultured midfielder had swapped passes with Craig Mackail-Smith who was easily the outstanding player on the field.
But Posh should have put the contest out of Northampton's reach as Mark Hughes needed to block a goalbound shot by Mackail-Smith after a mix-up between the defender and Chris Dunn, while the home goalkeeper twice saved from Chris Whelpdale inside a minute.
At the other end, Karl Hawley made an impressive debut following his loan move from Preston this week and saw a header deflected wide and caused a constant problem to the visitors.
Northampton could have levelled in the 40th minute when Luke Guttridge rattled the crossbar with a 30-yarder, but it was Guttridge who teed up Colin Larkin two minutes later, allowing the forward to bury a low shot past Joe Lewis.
Peterborough created the bulk of the chances in the second half but were guilty of some slipshod finishing.
In the 56th minute, Ryan Gilligan blocked a goalbound Mackail-Smith shot while goal hero Boyd blotted his copybook by missing the target two minutes later.
But the biggest howler arrived 13 minutes from time when Mackail-Smith carved open the home defence down the right and delivered an excellent cross for Scott Rendell only for the former Cambridge to screw badly wide.