Appleton's men saw off the Cherries 4-3 in the shoot-out after the first-round tie ended 2-2. Shaun MacDonald's double sandwiched Luke Rodgers' leveller before Brian Howard's 25-yard strike sent the game to penalties, and Lee Barnard and Marc Pugh missed from the spot to send Pompey through.
Appleton was delighted with the boost generated by his side's first win of the season, but he insists the club's off-field woes continue to prove a serious distraction to his makeshift side of loanees and players on short-term deals.
He said: "I just get the impression that everything that's going on off-the-field is getting to people at the minute. It's getting to everyone.
"It's getting to the point now where everyone is frustrated and you can see everyone is that little bit anxious.
It as if they feel, 'Would someone please take us out of administration and then we can all look forward to a brighter future'."