Matt Fryatt got the Tigers' 66th-minute opener from the penalty spot on his home league debut after the referee adjudged the striker's shirt had been pulled in the box by Stephen Foster, who was shown a straight card.
The Tykes' Jason Shackell was then also dismissed for a second booking by official Peter Quinn, before Robert Koren netted in injury time to secure the win for the Tigers.
Pearson said: "We have to recognise it is a very important part of the season for us and the next month or so is going to be vital. We're only six points off the play-offs, but it's still tight below us.
"We have made a bit of progress and if we go into games with the belief that we're building then I think we've got a chance with the additions we've made."