The Spireites made the perfect start to the League Two clash at Gigg Lane when Aaron Downes gave them a second-minute lead, before top scorer Jack Lester scored a superb second goal in the 40th minute.
Fourth-placed Bury were handed a lifeline when Gregor Robertson was sent off just two minutes later and although Andy Bishop halved the host's deficit with a 60th-minute penalty, Chesterfield put in a dogged second-half display to claim all three points.
"I want to pay tribute to the players because they were all excellent," said Richardson. "Bury did their utmost to break through. They tried to play through us, over us, round us, and there was a real colossal effort put in by our players and I am very proud of that performance.
"Bury probably had more possession in the first half but we certainly looked the most likely to score.
"We went in happy with a 2-0 lead but disappointed we lost Gregor for what looked to me like a yellow card. That meant we had to face the whole of the second half with 10 men and credit to the players because they defended ever so well."