The Spireites lost the first round tie 2-1 but many inside Elland Road on Wednesday night agreed that the League Two team were the better side.
Summer signing Eoin Doyle headed them in front after 19 minutes but two goals in four minutes from Michael Brown and Dominic Poleon restored some pride for the Championship club.
Chesterfield have now fallen at the first hurdle in the League Cup in the last seven seasons, but Richardson said: "We came here to give a good account of ourselves and we did that. The result was disappointing but there were a lot of positives to come out of the game.
"We are trying to gel as a team and make small steps together. We controlled a good proportion of the game, which breeds confidence. Our young players will come on by leaps and bounds. Now we have to raise our game against Cheltenham on Saturday."
Gary Roberts, Chesterfield's most effective player, felt he should have had a 50th-minute penalty when he was brought down by Zac Thompson but referee Carl Boyeson ruled the offence was outside the box.
Richardson added: "I thought it was a penalty but I would have to see it again to be sure. The officials have a tough job in those circumstances."