Chesterfield scored in the fouth minute when Gary Roberts crossed low to the far post, where Tendayi Darikwa fired past Southend keeper Dan Bentley, but the visitors equalised eight minutes later when Marc Laird headed in Ben Coker's corner.
Drew Talbot headed against the top of the Southend crossbar in the 52nd minute, but the visitors had a great chance to settle it in the 79th minute.
Sam Hird barged over Barry Corr to concede a penalty, but the striker's spot kick was superbly saved by Lee only for Hurst to finish it in the 85th minute from 12 yards after great work by Straker.
Penney said: "I thought from minute one despite conceding we were decent and it was completely against the run of play that they scored.
"But there's a little bit more character and resolve about us and we haven't got a soft underbelly if things go against us.
"In adversity we seem to galvanise ourselves as a team and step up to the plate.
"There's a real togetherness among the players now and you can see there's a great spirit.
"We got the goals and could have scored four with the penalty, but we are delighted to be in the next round.
"We pressed ever so well and didn't give them a second. There is a real desire and will to win about these players."