It took the Cobblers just seven minutes to breach the best away defence in the league when Robinson headed home Chris Hackett's delivery.
Robinson almost doubled Northampton's advantage moments later when his shot was blocked by Rob Atkinson and Hackett went close with a free-kick from just outside the box.
Hackett also saw another free-kick saved by Scott Davies and Adebayo Akinfenwa was off target with a header as the Cobblers dominated the opening exchanges.
Fleetwood's first chance fell to Junior Brown who headed wide from Damien Johnson's cross and just before the break David Ball fired past the post. But Robinson doubled Northampton's advantage four minutes before half time when he beat Davies with a low drive from the edge of the box.
Hackett went close with another effort in first half stoppage-time and just after the restart Akinfenwa's header was saved by Davies. In between Brown saw his volley saved by Lee Nicholls before the lead was increased in the 53rd minute when Akinfenwa set up Clive Platt who beat Davies from close range.
Alex Marrow and Conor McLaughlin had goalbound efforts blocked as Fleetwood tried to get back into the contest but it was not until two minutes into stoppage-time that Jamie Allen grabbed a consolation, beating Nicholls with a low drive.
