The Bees, who beat Bristol City in the previous round, hit Plymouth within 21 seconds of the second half as Grazioli beat a non-existent offside trap before chipping sublimely over keeper Luke McCormick.
It was the third great goal of the match after Barnet had taken an 11th minute lead through Simon King. He connected perfectly with his volley from the edge of the box after the Pilgrims had failed to fully clear a corner to send the ball crashing into the top corner of the net.
McCormick didn't even see it, yet alone react, and the Bees were good value for their lead. They could even have extended their advantage soon afterwards but King wasted another good opportunity with a free header.
However, the lead only lasted eight minutes as Akos Buzsaky equalised with a goal of pure quality from 25 yards. His powerful run had won the free-kick in the first place before he curled a shot beautifully over the wall and beyond the despairing dive of Scott Tynan.
But Championship outfit Plymouth failed to switch on at the start of the second half and paid the ultimate price. The offside trap failed completely and Grazioli made no mistake with a sweet chip that stunned the visitors.
It could have been even worse for Plymouth as Grazioli somehow missed the target with a header from just six yards as Barnet threatened to score with every meaningful attack.
Plymouth, meanwhile, were a massive disappointment under caretaker manager Jocky Scott and they created few chances to grind out another equaliser until the last few minutes, but Barnet held on for a famous victory.