After Matty Brown's first Football League goal, Marc Richards' spot-kick put Chesterfield 2-0 up and Plymouth had chances to get back in the game only to find Tommy Lee in superb form.
He saved Reid's 64th minute penalty and made two other brilliant saves in an afternoon which saw strikers Armand Gnanduillet and Marc Richards suffer injuries and Brown was red-carded.
"I thought the players were terrific, they had to put their hands to the wall and push hard and in the second half when we went to 10 men," he said. "We gave the club everything in terms of work ethic and desire to see the game out.
"We didn't give Plymouth anything to feed off because I'm sure if they had got one goal we would have found it a lot harder.
"There are a lot of tough, strong sides in this division and it's hard to put runs together so I think the lads deserve a big pat on the back. They certainly came to the fore and I am very proud of them."