The Spireites triumphed in their replay with Billericay - winning the game 3-1 thanks to a hat-trick from Tom Denton.
Allen insisted last week that his squad were improving and he echoed that comment after last night's win.
“It’s another game unbeaten. After the disaster at the end of September, I referred to it a few times now September the 25th,” he told the club's official video service.
“I think there’s been a massive improvement in our group. I think we are getting better. Some of the young players are really stepping up into the team now and it should give them confidence.
“I’m pleased for the players. All the hard work and commitment that they’ve put in to get us through to the next round,” said the manager.
“I’m pleased for all of the supporters that were here tonight, and for all the supporters at home that would have been watching it eagerly.
“It’s a tough place to come and play on a waterlogged pitch. They’re a good side. At the moment they are second in their league."
Chesterfield now face League Two Grimsby Town in the Second Round on Saturday December 1.