Victory would have moved Stanley to within two points of the League Two play-off places and Dean Winnard gave them the ideal start, heading home Charlie Barnett's corner after 78 seconds.
But the Reds could not build on that early breakthrough, with Daniel Nardiello levelling moments later and then firing the winner for Rotherham early in the second half.
"We lacked a spark and sometimes you've got to hold your hands up and say you weren't good enough," said Richardson.
"Things didn't come off for us and we didn't create a lot but they didn't either. Two mistakes proved costly.
"The lads are disappointed they haven't won, which shows how far we have come as a club, and all we can do now is look forward to the FA Cup on Saturday against Oxford which is a massive game for the club."