Cobblers enhanced their chances of reaching the play-offs against a Plymouth side hovering just above the relegation zone courtesy of Cole's second-half spillage, which allowed Platt to prod home from close range.
Plymouth started brightly, Maxime Blanchard tested Lee Nicholls who also denied Jason Banton, while Reuben Reid had a goalbound shot blocked and Guy Branston headed wide.
Northampton finally began to forge some openings - Roy O'Donovan saw his effort turned away by Cole, who also kept out Ben Harding's free-kick and Chris Hackett fired wide from the edge of the box.
Former Cobbler Anthony Charles forced Nicholls into action again just before the break but Northampton stepped up the tempo after the restart - O'Donovan heading wide, while Clarke Carlisle was also off target with a header of his own.
Adebayo Akinfenwa was introduced as the promotion chasers sought to make the breakthrough and it was the top scorer who was instrumental in Northampton doing just that in the 72nd minute. Akinfenwa forced Cole into a decent save but - from Harding's resulting corner - the Plymouth keeper could only help the ball into the path of Platt, who made no mistake in hitting the back of the net.
Charles missed a late chance to earn a point when he headed wide from Barton's corner but substitute Jake Robinson missed a great chance to make the game safe for Cobblers when he shot over.