UPDATE: Rowling has apparently just addressed this on Twitter.
Good story, but it isn't true! @stephenfry wasn't condescending, he was completely lovely, and that phrase isn't in 4 books! #factsmatter
Original answer below.
Almost, but not quite
Book 4: "Harry pocketed it"
He pulled Sirius’s reply off Pigwidgeon’s leg, Harry pocketed it, and they hurried back to Gryffindor Tower to read it. (Chapter 23)
Book 5: "pocketed it"
'Right-o,' Fred said brightly, spraying the Doxy quickly in the face so that it fainted, but the moment Mrs. Weasley's back was turned he pocketed it with a wink. (Chapter 6)
Book 6: "pocketed it"
Not long after this, Hagrid became tearful again and pressed the whole unicorn tail upon Slughorn, who pocketed it with cries of, “To friendship! To generosity! To ten Galleons a hair!” (chapter 22)
Book 7: No pocketing :-(
Note: While the question was only asking about the accuracy of the finale part, it should be noted that the phrase Harry pocket it is not found anywhere in The Prisoner of Azkaban, nor for that matter is the word pocketed.
Bonus: At the OP's request, I am adding a list of all of the places that the word pocketed can be found in Harry Potter Canon.
Harry pocketed it - 4 times
"Better get the cloak," Ron muttered, as Lee Jordan finally left, stretching and yawning. Harry ran upstairs to their dark dormitory. He putted out the cloak and then his eyes fell on the flute Hagrid had given him for Christmas. He pocketed it to use on Fluffy -- he didn't feel much like singing.
"Well, it's not much use to you," said Ron. He dropped his voice. "Fifty points if you can get it through Myrtle's nose."
Harry, however, pocketed it.
Mr. Diggory handed Harry his wand and Harry pocketed it. (GF9)
He pulled Sirius’s reply off Pigwidgeon’s leg, Harry pocketed it, and they hurried back to Gryffindor Tower to read it. (GF23)
Pocketed it - 2 times
'Right-o,' Fred said brightly, spraying the Doxy quickly in the face so that it fainted, but the moment Mrs. Weasley's back was turned he pocketed it with a wink. (OP6)
Not long after this, Hagrid became tearful again and pressed the whole unicorn tail upon Slughorn, who pocketed it with cries of, “To friendship! To generosity! To ten Galleons a hair!” (HBP22)
Pocketed - 4 times
"We're going to talk now," said Riddle, still smiling broadly, and he pocketed Harry's wand.
Each wore an expression of utmost contempt, yet the unexpected entrance of so many witnesses seemed to have brought them to their senses. Snape pocketed his wand(, turned on his heel)1 and swept back across the kitchen, passing the Weasleys without comment. At the door he looked back. (OP24)
He looked at Ron. "You saved everything because you felt lucky. You did it all yourself."
He pocketed the potion again. (HBP14)
Then he returned to the Gaunt hovel, performed the complex bit of magic that would implant a false memory in his uncle's mind, laid Morfin's wand beside its unconscious owner, pocketed the ancient ring he wore, and departed." (HBP17)
Philosopher's Stone - 1 occurrence, He [harry] pocketed it
Chamber of Secrets - 2 occurrences, including Harry(, however,) pocketed it.
Prisoner of Azkaban - No occurrences
Goblet of Fire - 2 occurrences, both of which are Harry pocketed it.
Order of the Phoenix - 2 occurrences, including pocketed it.
Half-Blood Prince - 3 occurrences, including pocketed it
Deathly Hallows - no occurrences
Supplementary works - no occurrences
Interviews - no occurrences
Pottermore - no occurrences
1 ", turned on his heel" was removed in later editions.