The Northamptonshire part-timers marked their first-ever appearance in the first round by frustrating Chesterfield until Roberts pounced in the 69th minute, while Ryan struck late on.
The Southern League Division One Central side showed plenty of confidence 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.
Ross Oulton then forced Chesterfield keeper 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 captain Liam Dolman and having midfielder Mykel Beckley stretchered off, were frustrating the home side until Chesterfield finally broke through in the 69th minute.
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.