The Cobblers laid siege to the Shots goal from the start of Saturday's League Two encounter but the visitors withstood the early barrage.
Goalkeeper Jamie Young kept out Chris Hackett's header and Ishmel Demontagnac's close-range effort was blocked for a corner from which Kelvin Langmead saw his shot hacked away.
Aldershot went close when Danny Hylton headed wide from Kieron Cadogan's cross.
Clarke Carlisle should have given the hosts the lead when he headed over from Hackett's free-kick.
It took just four minutes for Robinson to break the deadlock after coming on at half-time, racing on to Ben Tozer's pass to beat Young.
Robinson was denied by Young moments later before Hylton missed a glorious chance to equalise when he fired wide at the far post.
Daniel Lopez was denied by Cobblers keeper Alex Nicholls after racing clear while at the other end Young tipped over Hackett's dipping free-kick.
But Platt wrapped up the points for Northampton deep into stoppage-time when he lobbed the advancing Young after latching onto a long clearance.