Scott Hogan's 36th-minute goal proved decisive with the Shakers unable to make numerical advantage count after Peter Vincenti's dismissal for a second bookable offence soon after.
"It was a very big test of character, probably more so from the decision of Peter getting sent off," Beech said.
"It was a foul, as was the first one, and once the referee has made his decision we had to respond to it. We re-organised at half time and the lads worked really hard, stuck to the plan and waited to see what Bury would do. We had the lead and it was a case of challenging them to beat us rather than beating ourselves.
"Sometimes it can lure the opposition into a false sense of security having an extra man and it certainly poses the question 'how good are you with the ball?'
"We just allowed their two centre-halves to have the ball, go with two banks of four behind the ball and just look to engage around the halfway line to try and nullify any passing opportunities through the centre of our team."
Meanwhile, Bury boss Kevin Blackwell hit out at at Rochdale's rough-house tactics along with referee Carl Boyeson's policing of the game.
"Rochdale were lucky to have 10 men on in the end because some of their tackling was outrageous," Blackwell said.
"Keith Hill had a go at the referee at half-time, screaming in his face, and for me that showed in the second half. Hilly has done his job, I'm just disappointed with the officials.
"There was a point at half-time when I was worried we didn't have enough ice in the dressing room. It's up to the referee to sort that out, otherwise it becomes a free-for-all.
"But having said that we didn't do enough with the ball and that concerns me. As good as our strikers have been lately, they weren't good enough on the day.
"It's our own fault because we had enough possession. Our frontmen were nowhere near bright enough and I gave someone a job to do at a corner and he didn't do his job and the ball ends up in the back of the net."