Darren Powell headed Palace into the lead in the 31st minute before Blair Sturrock levelled seven minutes later.
The Eagles dominated the second half but had to wait until the 113th minute before Johnson clinched the tie.
After a lack-lustre opening Darren Powell eventually brought some cheer from the empty terraces wtih a well-placed header in the 31st minute.
Danny Granville's inswinging free kick found Dele Adebola at the far post and he comfortably headed across goal for Powell to meet it with a well-placed header which nestled in the net by the right post.
The Pilgrims hit back immediately and Sturrock was guilty of an incredible miss with an open goal in front of him.
But the manager's son made no mistake seven minutes before the break with a first-time shot ten yards out after great work by Osvaldo Lopes who supplied the perfect cross.
After the break Martin Phillips wasted a good chance with a clear header but Palace gradually got a grip and the chances began to flow their way.
Wayne Routledge, only 17-years-old and making his first start, caused all sorts of problems for the Pilgrims.
Tommy Black fired wide from Dougie Freeman's pass and Granville shot across the face of goal.
Palace kept plugging away and Granville, Freedman and Black were all off target to ensure extra time.
Still Palace could not find the net as Romain Larrieu saved twice to deny Black in the space of two minutes.
Credit to Plymouth as they stuck to their task, relying on the counter-attack, but their efforts were all in vain. Seven minutes from the end Freedman sprayed another fine ball into the path of Granville who crossed for Johnson to steer past Larrieu for a deserved victory