Alessandra had twice gone close in the final five minutes before latching on to Reuben Reid's throughball and slotting past Cobblers goalkeeper Matt Duke.
Plymouth's Curtis Nelson could easily have scored at both ends in the first half.
The young defender headed the ball on to his own post in the 21st minute following Chris Hackett's cross before forcing a crucial save from Duke with a first-time strike direct from Alessandra's corner.
Argyle 'keeper Luke McCormick made a superb save to palm over Luke Norris' angled strike on the run at the start of the second half.
Duke did well to turn Reid's goalbound 20-yard shot around a post after it took a late deflection.
Substitute Paul Hayes sent the ball across the face of Northampton's goal and although Alessandra lunged in to get a touch it was not enough.
Seconds later Alessandra sent Duke full stretch with a measured 20-yard strike.