Trailing 1-0 from the first leg, the Robins were handed a chance to level the tie with the player's 14th-minute spot-kick but his effort was saved by Cobblers goalkeeper Lee Nicholls.
Mark Yates' side went on to lose 2-0 on aggregate and Pack was left deeply upset with his last game for the club.
"I feel like I've let everyone down and it's hard to take at the minute," he told BBC Gloucestershire. "I'm distraught and it's going to take me a while to get over it.
"I've scored three penalties this year and I was wondering whether to change my mind but I stuck to my guns and the keeper pulled off the save.
"I love this sport but right now it's a ruthless one."
Pack, 22, came close to leaving the club in January and is expected to leave this summer.
"I've had a great two seasons but it's not the way I would have liked to have finished," Pack added
"I want to thank everyone for the time I've had here. I've grown up here and as a player I've matured.
"It will be a time that will always be close to my heart throughout my career. The fans have been great to me since day one and I can't thank them enough for that."
- Sports & Recreation
- Lee Nicholls