Stanley dominated the League Two contest for most of the first hour, but could find no answer to Billy Kee's early goal against his former club.
Beattie might have made all the difference but had to be pulled from the team after picking up an injury in the warm-up.
Richardson said: "I don't think we deserved to lose the game in respect of our good possession. We got into good areas and the only thing that was lacking was that we needed to be a bit more clinical.
"Unfortunately, you don't win football games with possession and you don't win them with chances on goal. You win games by putting the ball in the back of the net - and Burton beat us on that score.
"Their goal came against the run of play, but you have to deal with setbacks like that, but all we did was knock on the door without managing to score.
"It was disappointing that we worked so hard to get in good areas without making the most of them.
"Obviously James Beattie was a big miss for us, but he felt something in the warm-up and I decided to pull him out of the team rather than run the risk of him possibly missing more games."
It took Kee only 10 minutes to open the scoring from 10 yards after a crisp exchange of passes with fellow striker Calvin Zola inside the box.
Accrington looked much the brighter team in the first half, but Burton had chances to increase their lead.
