The Northamptonshire part-timers held the home side until the 69th minute when Gary Roberts pounced, but Chesterfield's place in the second round was only sealed by Jimmy Ryan's deflected shot two minutes from time.
Richardson said: "The important thing is we are through to the second round, but we have to compliment Daventry on the endeavour and performance they put in.
"Their manager said it was the biggest day in the club's history so they are going to come and have a go which is exactly what we saw.
"We did think in the end we would wear them down and that the fitness levels and quality would show which they did.
"When teams haven't got anything to lose or fear they are a dangerous animal and when you have everything to lose there is a bit of anxiety.
"But you believe in what you do and the ultimate goal was to be in the next round.
"I think the club has come on leaps and bounds even in the short time I've been here and it's important that we continue to believe in what we are doing and do it well."
Daventry manager Darran Foster admitted he was "gobsmacked" after his team came close to causing an upset.
"I'm gobsmacked," he said. "The players have been wonderful today and I'm so proud of them. They came up here against a team that's flying and I felt we shaded the first half."
The Southern League Division One Central side showed plenty of confidence from the first whistle and had a chance to take the lead in the 34th minute when Tom Lorraine got in on the right, but his shot went wide off a defender and Ross Oulton forced Tommy Lee into a diving save three minutes before half time.
Daventry 'keeper Richard Morris denied Gary McSheffrey in the 50th minute but Daventry, despite losing their captain Liam Dolman and having midfielder Mykel Beckley stretchered off, were frustrating the home side until Chesterfield finally broke through.
A free kick to the back post found Roberts who guided the ball back across and inside Morris's left hand post and Ryan ended Daventry's dream with a deflected 25 yard shot in the 88th minute.
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- Jimmy Ryan