Chairman Mitchell was responding after Southampton manager Nigel Adkins threw down the gauntlet to Cherries ahead of Saturday's much-anticipated Dean Court derby.
Speaking following Saints' 3-0 win over Yeovil on Tuesday, Adkins challenged Cherries to match our intensity and said the visitors from Hampshire would be up for it.
Victory for Saints in the first league meeting between the clubs at Dean Court for more than 50 years would see them leapfrog Lee Bradbury's men in League One.
Some Cherries supporters are planning to hold aloft cardboard cups adored with Adkins' face, a sarcastic reference to Saints followers describing every match to their opponents as a cup final.
Despite being installed in pre-season as warm favourites for the title, Southampton's stuttering form has seen them become embroiled in a tense battle for promotion and trailing Cherries for much of the campaign.
If they think it is our cup final, it is very much theirs as well, said Mitchell. Who would have thought at the start of the season that we would be playing Southampton in a promotion six-pointer in mid-March
I spoke to someone quite high up at Southampton who watched us last season and they said we would suffer badly in this division and probably come straight back down.
We have already surpassed our expectations for the season and they still have theirs to achieve. I think it shows just how far we have come that we are in this position. We have been above them for most of the season and want to keep it that way.
We had an off-night at Exeter on Tuesday and Southampton had a good result against Yeovil. One result doesn't make a season though and we are second in the table on merit.
We deserve to be where we are and our home record is excellent. The pressure is on both teams because there is much at stake in this game. But we would have taken this scenario at the start of the season.
The result will go a long way to deciding who gets automatic promotion and the winner will be in prime position to go up. A win would see us open a gap on them so there is all to play for.
Mitchell added: We are now recognised as a force in this division and, in my opinion, we have got a Championship club here. It is only a matter of time until we get there. I always like to do things in a rush so want to get there this season. I am very proud of everybody at the club for getting us into this situation.
It is not just about money and the size of your stadium, it is about rising to the challenge and we have done this season. It is for us to grasp the season by the scruff of the neck and finish the job.