A crowd of 9,043 which included 1,283 visiting supporters saw Cherries and Huddersfield play out an entertaining 1-1 draw at Dean Court on Saturday.
The gate was just 38 more than for Cherries' final league game against Rochdale the previous week and showed a reduction of 843 home supporters.
It was the eighth highest attendance since the stadium was redeveloped in 2001 and almost 1,000 down on the record 10,008 against Southampton in March.
Chairman Mitchell told the Daily Echo: It was a big disappointment, especially when the boys deserved a full house for getting this far against the odds. I wanted every seat in the house sold and they weren't.
Despite banking a £20,000 broadcast fee from Sky Sports, Cherries missed out on valuable income after failing to sell out for what was arguably the biggest game to be staged at the venue in recent years.
Asked whether he had been happy with the club's ticketing policy, Mitchell replied: They were always available to buy. As late as Thursday and Friday, it had been publicised that tickets were still available.
On reflection, could we have done a bit more advertising to sell them To be honest, I didn't really expect to be in that situation. I thought they would sell but it wasn't to be.
Mitchell added: Lunchtime is always an odd time to play and, for me, it doesn't sit very well. Maybe some people felt the same.
With the game being shown on Sky, the benefits were huge for the players. It gave them a chance to star on the small screen. In that respect, it was great, particularly for those who played well.
To have money coming in from Sky for a mid-winter game is a bonus. But, as a club and for family football, on this occasion, we would have preferred the extra 1,000 people rather than the Sky money. There is no rhyme nor reason for it not being a full house.
Cherries are hoping to have a sizeable away following at the Galpharm Stadium for tomorrow's second leg after the club priced at cost coach travel. Leaving Dean Court at 11.45am, supporters will pay £25 per person and must purchase a match ticket at the same time.
Meanwhile, Mitchell confirmed the club would be spending around £12,000 on the pitch and surrounding area at Dean Court during the summer.
The plans include extensive work on the playing surface and laying asphalt on the area between the pitch and the stands.
He also said the home dressing room would be redecorated during the close season.