But he has big concerns about a growing injury crisis for his squad which now sees seven sidelined with long-term problems.
First-half goals from Sean McAllister and former Spireites Gary McSheffrey eased the Iron to victory.
"That's the first back-to-back victories for us this season and obviously its pleasing to be in the draw for the second round," said Wilcox.
"I can't praise my players highly enough for what they have produced in the last three or four games. They have done this club proud - and now we are going to have to ask them for another big effort at Bristol City on Saturday."
Scunthorpe may be forced to play winger Terry Hawkridge as an emergency full-back at Ashton Gate because of their injury situation.
The Iron lost centre-back Miles Addison after last weekend with a knee injury and full-back Luke O'Neill left the ground on crutches following the win against Chesterfield.
"In my 30 years in football, I have never quite known anything like it," groaned Wilcox.
"I know I keep ranting on about it, but we have now played nine games in the opening four weeks of the season.
"It's crazy and a big ask for clubs like us with a small squad.
"They are not just knocks or bruises, we have seven players out with long-term injuries. We really are down to the bare bones.
"We had just agreed to extend Miles' loan from Bournemouth but now he has gone back there for them to assess him and see if he needs an operation.
"He was just settling into the team and when he is fit again he is someone I would definitely want to try and bring back.
"Until the loan window opens again, we have to go with what we have got. And that means we might have to play young Terry Hawkridge at full-back at Bristol. I can't think of anyone else we have available.
"The injury list here just keeps growing - and that is a big concern."
Chesterfield boss Paul Cook was upbeat about the performance of some of the younger players he blooded into the team, but disappointed with the result.
"We gave some of them their debuts and I was pleased with the way they performed," he said.
"I would have loved to see them get a goal and get back into the game, but it didn't happen.
"The experience they have gained will be vital for them all.
"It's hard to enthuse too much though about a performance when we have been beaten.
"That is something we have got at this club at he moment. We do not like losing.
"We are disappointed to have gone out but we have just got now to look forward to the next match.
"We play Scunthorpe again in 10 days time in the league and hopefully we can get a different result."